research article orthogonally additive and...

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 415354 9 pageshttpdxdoiorg1011552013415354

Research ArticleOrthogonally Additive and Orthogonality PreservingHolomorphic Mappings between Clowast-Algebras

Jorge J Garceacutes1 Antonio M Peralta1 Daniele Puglisi2 and Mariacutea Isabel Ramiacuterez3

1 Departamento de Analisis Matematico Facultad de Ciencias Universidad de Granada 18071 Granada Spain2Dipartimento di Matematica e Informatica Universita di Catania 95125 Catania Italy3 Departamento de Algebra y Analisis Matematico Universidad de Almerıa 04120 Almerıa Spain

Correspondence should be addressed to Antonio M Peralta aperaltaugres

Received 1 October 2013 Accepted 27 November 2013

Academic Editor Ngai-Ching Wong

Copyright copy 2013 Jorge J Garces et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We study holomorphic maps between Clowast-algebras 119860 and 119861 when 119891 119861119860(0 984858) rarr 119861 is a holomorphic mapping whose Taylor

series at zero is uniformly converging in some open unit ball 119880 = 119861119860(0 120575) If we assume that 119891 is orthogonality preserving and

orthogonally additive on 119860119904119886cap 119880 and 119891(119880) contains an invertible element in 119861 then there exist a sequence (ℎ

119899) in 119861lowastlowast and Jordan

lowast-homomorphisms Θ Θ 119872(119860) rarr 119861lowastlowast such that 119891(119909) = sum

infin

119899=1ℎ119899

Θ(119886

119899

) = suminfin

119899=1Θ(119886

119899

)ℎ119899uniformly in 119886 isin 119880 When 119861 is abelian

the hypothesis of 119861 being unital and 119891(119880) cap inv(119861) = 0 can be relaxed to get the same statement

1 Introduction

The description of orthogonally additive 119899-homogeneouspolynomial on 119862(119870)-spaces and on general Clowast-algebrasdeveloped by Benyamini et al [1] Perez-Garcıa and Vil-lanueva [2] and Palazuelos et al [3] respectively (see also[4 5] [6 Section 3] and [7]) made functional analysts studyand explore orthogonally additive holomorphic functions on119862(119870)-spaces (see [8 9]) and subsequently on general Clowast-algebras (cf [10])

We recall that a mapping 119891 from a Clowast-algebra 119860 into aBanach space 119861 is said to be orthogonally additive on a subset119880 sube 119860 if for every 119886 119887 in 119880 with 119886 perp 119887 and 119886 + 119887 isin 119880

we have 119891(119886 + 119887) = 119891(119886) + 119891(119887) where elements 119886 119887 in119860 are said to be orthogonal (denoted by 119886 perp 119887) whenever119886119887

lowast

= 119887lowast

119886 = 0 We will say that 119891 is additive on elementshaving zero product if for every 119886 119887 in 119860 with 119886119887 = 0 wehave 119891(119886 + 119887) = 119891(119886) + 119891(119887) Having this terminology inmind the description of all 119899-homogeneous polynomials ona general Clowast-algebra 119860 which are orthogonally additive onthe self-adjoint part 119860

119904119886 of 119860 reads as follows (see Section 2

for concrete definitions not explained here)

Theorem 1 (see [3]) Let 119860 be a Clowast-algebra and 119861 a Banachspace 119899 isin N and let 119875 119860 rarr 119861 be an 119899-homogeneouspolynomial The following statements are equivalent

(a) There exists a bounded linear operator 119879 119860 rarr 119861

satisfying

119875 (119886) = 119879 (119886119899

) (1)

for every 119886 isin 119860 and 119875 le 119879 le 2119875(b) 119875 is additive on elements having zero products(c) 119875 is orthogonally additive on 119860

119904119886

The task of replacing 119899-homogeneous polynomials bypolynomials or by holomorphic functions involves a higherdifficulty For example as noticed by Carando et al [8Example 22] when 119870 denotes the closed unit disc in Cthere is no entire function Φ C rarr C such that themapping ℎ 119862(119870) rarr 119862(119870) ℎ(119891) = Φ ∘ 119891 factorizesall degree-2 orthogonally additive scalar polynomials over119862(119870) Furthermore similar arguments show that defining 119875

119862([0 1]) rarr C 119875(119891) = 119891(0) + 119891(1)2 we cannot find a triplet

2 Abstract and Applied Analysis

(Φ 1205721 120572

2) where Φ 119862[0 1] rarr C is a lowast-homomorphism

and 1205721 120572

2isin C satisfying that 119875(119891) = 120572

1Φ(119891) + 120572

2Φ(119891

2

) forevery 119891 isin 119862([0 1])

To avoid the difficulties commented above Carandoet al introduce a factorization through an 119871

1(120583) space

More concretely for each compact Hausdorff space 119870 aholomorphic mapping of bounded type 119891 119862(119870) rarr C

is orthogonally additive if and only if there exist a Borelregular measure 120583 on 119870 a sequence (119892

119896)119896sube 119871

1(120583) and a

holomorphic function of bounded type ℎ 119862(119870) rarr 1198711(120583)

such that ℎ(119886) = suminfin

119896=0119892119896119886119896 and

119891 (119886) = int

119870

ℎ (119886) 119889120583 (2)

for every 119886 isin 119862(119870) (cf [8 Theorem 33])When 119862(119870) is replaced with a general Clowast-algebra 119860

a holomorphic function of bounded type 119891 119860 rarr C

is orthogonally additive on 119860119904119886

if and only if there exist apositive functional 120593 in 119860

lowast a sequence (120595119899) in 119871

1(119860

lowastlowast

120593)and a power series holomorphic function ℎ in H

119887(119860 119860

lowast

)

such that

ℎ (119886) =

infin

sum

119896=1

120595119896sdot 119886

119896

119891 (119886) = ⟨1119860lowastlowast ℎ (119886)⟩ = int ℎ (119886) 119889120593

(3)

for every 119886 in 119860 where 1119860lowastlowast denotes the unit element in 119860lowastlowast

and 1198711(119860

lowastlowast

120593) is a noncommutative 1198711-space (cf [10])

A very recent contribution due to Bu et al [11] showsthat for holomorphicmappings between119862(119870) spaces we canavoid the factorization through an 119871

1(120583)-space by imposing

additional hypothesis Before stating the detailed result wewill set down some definitions

Let 119860 and 119861 be Clowast-algebras When 119891 119880 sube 119860 rarr 119861 is amap and the condition

119886 perp 119887 997904rArr 119891 (119886) perp 119891 (119887)

(resp 119886119887 = 0 997904rArr 119891 (119886) 119891 (119887) = 0)

(4)

holds for every 119886 119887 isin 119880 we will say that 119891 preserves orthogo-nality or it is orthogonality preserving (resp 119891 preserves zeroproducts) on 119880 In the case 119860 = 119880 we will simply say that 119891is orthogonality preserving (resp 119891 preserves zero products)Orthogonality preserving bounded linear maps between Clowast-algebras were completely described in [12 Theorem 17] (see[6] for completeness)

The following Banach-Stone type theorem for zero prod-uct preserving or orthogonality preserving holomorphicfunctions between 119862

0(119871) spaces is established by Bu et al in

[11 Theorem 34]

Theorem 2 (see [11]) Let 1198711and 119871

2be locally compact

Hausdorff spaces and let 119891 1198611198620(1198711)(0 119903) rarr 119862

0(119871

2) be a

bounded orthogonally additive holomorphic function If 119891 iszero product preserving or orthogonality preserving then thereexist a sequence (O

119899) of open subsets of 119871

2 a sequence (ℎ

119899)

of bounded functions from 1198712cup infin into C and a mapping

120593 1198712rarr 119871

1such that for each natural 119899 the function ℎ

119899is

continuous and nonvanishing on O119899and

119891 (119886) (119905) =

infin

sum

119899=1

ℎ119899(119905) (119886 (120593 (119905)))

119899

(119905 isin 1198712) (5)

uniformly in 119886 isin 1198611198620(1198711)(0 119903)

The study developed by Bu et al is restricted to com-mutative Clowast-algebras or to orthogonality preserving andorthogonally additive 119899-homogeneous polynomials betweengeneral Clowast-algebras The aim of this paper is to extend theirstudy to holomorphic maps between general Clowast-algebrasIn Section 4 we determine the form of every orthogonalitypreserving and orthogonally additive holomorphic functionfrom a general Clowast-algebra into a commutative Clowast-algebra(see Theorem 16)

In the wider setting of holomorphic mappings betweengeneral Clowast-algebras we prove the following let 119860 and 119861

be Clowast-algebras with 119861 unital and let 119891 119861119860(0 984858) rarr 119861

be a holomorphic mapping whose Taylor series at zero isuniformly converging in some open unit ball 119880 = 119861

119860(0 120575)

Suppose 119891 is orthogonality preserving and orthogonallyadditive on119860

119904119886cap119880 and 119891(119880) contains an invertible element

Then there exist a sequence (ℎ119899) in 119861

lowastlowast and Jordan lowast-homomorphisms Θ Θ 119872(119860) rarr 119861

lowastlowast such that

119891 (119909) =

infin

sum

119899=1

ℎ119899

Θ (119886

119899

) =

infin

sum

119899=1

Θ(119886119899

) ℎ119899 (6)

uniformly in 119886 isin 119880 (see Theorem 18)The main tool to establish our main results is a newfan-

gled investigation on orthogonality preserving pairs of oper-ators between Clowast-algebras developed in Section 3 Amongthe novelties presented in Section 3 we find an innovatingalternative characterization of orthogonality preserving oper-ators between Clowast-algebras which complements the originalone established in [12] (see Proposition 14) Orthogonalitypreserving pairs of operators are also valid to determineorthogonality preserving operators and orthomorphisms orlocal operators on Clowast-algebras in the sense employed byZaanen [13] and Johnson [14] respectively

2 Orthogonally Additive OrthogonalityPreserving and Holomorphic Mappingson Clowast-Algebras

Let 119883 and 119884 be Banach spaces Given a natural 119899 a(continuous) 119899-homogeneous polynomial 119875 from 119883 to 119884 isa mapping 119875 119883 rarr 119884 for which there is a (continuous)119899-linear symmetric operator 119860 119883 times sdot sdot sdot times 119883 rarr 119884 suchthat 119875(119909) = 119860(119909 119909) for every 119909 isin 119883 All polynomialsconsidered in this paper are assumed to be continuous Bya 0-homogeneous polynomial we mean a constant functionThe symbol P(

119899

119883119884) will denote the Banach space of allcontinuous 119899-homogeneous polynomials from 119883 to 119884 withnorm given by 119875 = sup

119909le1119875(119909)

Throughout the paper the word operator will alwaysstand for a bounded linear mapping

Abstract and Applied Analysis 3

We recall that given a domain 119880 in a complex Banachspace119883 (ie an open connected subset) a function119891 from119880

to another complex Banach space 119884 is said to be holomorphicif the Frechet derivative of 119891 at 119911

0exists for every point 119911

0in

119880 It is known that 119891 is holomorphic in 119880 if and only if foreach 119911

0isin 119883 there exists a sequence (119875

119896(119911

0))119896of polynomials

from 119883 into 119884 where each 119875119896(119911

0) is 119896-homogeneous and a

neighborhood 1198811199110

of 1199110such that the series

infin

sum

119896=0

119875119896(119911

0) (119910 minus 119911

0) (7)

converges uniformly to119891(119910) for every119910 isin 1198811199110

Homogeneouspolynomials on a Clowast-algebra 119860 constitute the most basicexamples of holomorphic functions on 119860 A holomorphicfunction 119891 119883 rarr 119884 is said to be of bounded type if it isbounded on all bounded subsets of 119883 in this case its Taylorseries at zero 119891 = sum

infin

119896=0119875119896 has infinite radius of uniform

convergence that is lim sup119896rarrinfin

1198751198961119896

= 0 (compare [15Section 62] see also [16])

Suppose119891 119861119883(0 120575) rarr 119884 is a holomorphic function and

let 119891 = suminfin

119896=0119875119896be its Taylor series at zero which is assumed

to be uniformly convergent in 119880 = 119861119883(0 120575) Given 120593 isin 119884lowast it

follows from Cauchyrsquos integral formula that for each 119886 isin 119880we have

120593119875119899(119886) =

1

2120587119894

int

120574

120593119891 (120582119886)

120582119899+1

119889120582 (8)

where 120574 is the circle forming the boundary of a disc in thecomplex plane 119863C(0 1199031) taken counterclockwise such that119886 + 119863C(0 1199031)119886 sube 119880 We refer to [15] for the basic facts anddefinitions used in this paper

In this section we will study orthogonally additiveorthogonality preserving and holomorphic mappingsbetween Clowast-algebras We begin with an observation whichcan be directly derived from Cauchyrsquos integral formulaThe statement in the next lemma was originally stated byCarando et al in [8 Lemma 11] (see also [10 Lemma 3])

Lemma 3 Let 119891 119861119860(0 984858) rarr 119861 be a holomorphic mapping

where 119860 is a Clowast-algebra and 119861 is a complex Banach spaceand let 119891 = sum

infin

119896=0119875119896be its Taylor series at zero which is

uniformly converging in 119880 = 119861119860(0 120575) Then the mapping 119891

is orthogonally additive on 119880 (resp orthogonally additive on119860

119904119886cap 119880 or additive on elements having zero product in 119880) if

and only if all the 119875119896rsquos satisfy the same property In such a case

1198750= 0

We recall that a functional 120593 in the dual of a Clowast-algebra119860is symmetric when 120593(119886) isin R for every 119886 isin 119860

119904119886 Reciprocally

if 120593(119887) isin R for every symmetric functional 120593 isin 119860lowast the

element 119887 lies in 119860119904119886 Having this in mind our next lemma

also is a direct consequence of Cauchyrsquos integral formula andthe power series expansion of 119891 A mapping 119891 119860 rarr 119861

between Clowast-algebras is called symmetric whenever 119891(119860119904119886) sube

119861119904119886 or equivalently 119891(119886) = 119891(119886)lowast whenever 119886 isin 119860

119904119886


where 119860 and 119861 are Clowast-algebras and let 119891 = suminfin

119896=0119875119896be its

Taylor series at zero which is uniformly converging in 119880 =

119861119860(0 120575) Then the mapping 119891 is symmetric on119880 (ie 119891(119860

119904119886cap

119880) sube 119861119904119886) if and only if 119875

119896is symmetric (ie 119875

119896(119860

119904119886) sube 119861

119904119886)

for every 119896 isin N cup 0

Definition 5 Let 119878 119879 119860 rarr 119861 be a couple of mappingsbetween two Clowast-algebras One will say that the pair (119878 119879) isorthogonality preserving on a subset 119880 sube 119860 if 119878(119886) perp 119879(119887)

whenever 119886 perp 119887 in119880 When 119886119887 = 0 in119880 implies 119878(119886)119879(119887) = 0in 119861 we will say that (119878 119879) preserves zero products on 119880

We observe that a mapping 119879 119860 rarr 119861 is orthogonalitypreserving in the usual sense if and only if the pair (119879 119879)is orthogonality preserving We also notice that (119878 119879) isorthogonality preserving (on 119860

119904119886) if and only if (119879 119878) is

orthogonality preserving (on 119860119904119886)

Our next result assures that the 119899-homogeneous poly-nomials appearing in the Taylor series of an orthogonalitypreserving holomorphic mapping between Clowast-algebras arepairwise orthogonality preserving

Proposition 6 Let 119891 119861119860(0 984858) rarr 119861 be a holomorphic

mapping where 119860 and 119861 are Clowast-algebras and let 119891 = suminfin

119896=0119875119896

be its Taylor series at zero which is uniformly converging in119880 = 119861

119860(0 120575) The following statements hold

(a) The mapping 119891 is orthogonally preserving on 119880 (resporthogonally preserving on119860

119904119886cap119880) if and only if119875

0= 0

and the pair (119875119899 119875

119898) is orthogonality preserving (resp

orthogonally preserving on 119860119904119886) for every 119899119898 isin N

(b) Themapping119891 preserves zero products on119880 if and onlyif 119875

0= 0 and for every 119899119898 isin N the pair (119875

119899 119875

119898)

preserves zero products

Proof (a) The ldquoif rdquo implication is clear To prove the ldquoonly if rdquoimplication let us fix 119886 119887 isin 119880 with 119886 perp 119887 Let us find twopositive scalars 119903 119862 such that 119886 119887 isin 119861(0 119903) and 119891(119909) le

119862 for every 119909 isin 119861(0 119903) sub 119861(0 119903) sube 119880 From the Cauchyestimates we have 119875

119898 le 119862119903

119898 for every 119898 isin N cup 0 Byhypothesis 119891(119905119886) perp 119891(119905119887) for every 119903 gt 119905 gt 0 hence

1198750(119905119886) 119875

0(119905119887)

lowast

+ 1198750(119905119886)(

infin

sum

119896=1

119875119896(119905119887))

lowast

+ (

infin

sum

119896=1

119875119896(119905119886))(

infin

sum

119896=0

119875119896(119905119887))

lowast

= 0

(9)

and by homogeneity

1198750(119886) 119875

0(119887)

lowast

= minus 1198750(119886)(

infin

sum

119896=1

119905119896

119875119896(119887))

lowast

+ (

infin

sum

119896=1

119905119896

119875119896(119886))(

infin

sum

119896=0

119905119896

119875119896(119887))

lowast

(10)

Letting 119905 rarr 0 we have1198750(119886)119875

0(119887)

lowast

= 0 In particular1198750= 0


We will prove by induction on 119899 that the pair (119875119895 119875

119896) is

orthogonality preserving on 119880 for every 1 le 119895 119896 le 119899 Since119891(119905119886)119891(119905119887)

lowast

= 0 we also deduce that

1198751(119905119886) 119875

1(119905119887)

lowast

+ 1198751(119905119886)(

infin

sum

119896=2

119875119896(119905119887))

lowast

+ (

infin

sum

119896=2

119875119896(119905119886))(

infin

sum

119896=1

119875119896(119905119887))

lowast

= 0

(11)

for every (min119886 119887)119903 gt 119905 gt 0 which implies that

1199052

1198751(119886) 119875

1(119887)

lowast

= minus 1199051198751(119886)(

infin

sum

119896=2

119905119896

119875119896(119887))

lowast

minus (

infin

sum

119896=2

119905119896

119875119896(119886))(

infin

sum

119896=1

119905119896

119875119896(119887))

lowast

(12)

for every (min119886 119887)119903 gt 119905 gt 0 and hence

10038171003817100381710038171198751(119886) 119875

1(119887)

lowast1003817100381710038171003817le 119905119862

10038171003817100381710038171198751(119886)

1003817100381710038171003817

infin

sum

119896=2

119887119896

119903119896

119905119896minus2

+ 1199051198622

(

infin

sum

119896=2

119886119896

119903119896

119905119896minus2

)(

infin

sum

119896=1

119887119896

119903119896

119905119896minus1

)

(13)

Taking limit in 119905 rarr 0 we get 1198751(119886)119875

1(119887)

lowast

= 0 Let us assumethat (119875

119895 119875

119896) is orthogonality preserving on 119880 for every 1 le

119895 119896 le 119899 Following the argument above we deduce that

1198751(119886) 119875

119899+1(119887)

lowast

+ 119875119899+1

(119886) 1198751(119887)

lowast

= minus1199051198751(119886)(

infin

sum

119895=119899+2

119905119895minus119899minus2

119875119895(119887))

lowast

minus 119905

119899

sum

119896=2

119905119896minus2

119875119896(119886)(

infin

sum

119895=119899+1

119905119895minus119899minus1

119875119895(119887))

lowast

minus 119905119875119899+1

(119886)(

infin

sum

119895=2

119905119895minus2

119875119895(119887))

lowast

minus 119905(

infin

sum

119896=119899+2

119905119896minus119899minus2

119875119896(119886))(

infin

sum

119895=1

119905119895minus1

119875119895(119887))

lowast

(14)

for every (min119886 119887)119903 gt |119905| gt 0 Taking limit in 119905 rarr 0we have

1198751(119886) 119875

119899+1(119887)

lowast

+ 119875119899+1

(119886) 1198751(119887)

lowast

= 0 (15)

Replacing 119886 with 119904119886 (119904 gt 0) we get

1199041198751(119886) 119875

119899+1(119887)

lowast

+ 119904119899+1

119875119899+1

(119886) 1198751(119887)

lowast

= 0 (16)

for every 119904 gt 0 which implies that

1198751(119886) 119875

119899+1(119887)

lowast

= 0 (17)

In a similar manner we prove that 119875119896(119886)119875

119899+1(119887)

lowast

= 0 forevery 1 le 119896 le 119899 + 1 The equalities 119875

119896(119887)

lowast

119875119895(119886) = 0 (1 le

119895 119896 le 119899 + 1) follow similarlyWe have shown that for each 119899119898 isin N 119875

119899(119886) perp 119875

119898(119887)

whenever 119886 119887 isin 119880 with 119886 perp 119887 Finally taking 119886 119887 isin 119860 with119886 perp 119887 we can find a positive 120588 such that 120588119886 120588119887 isin 119880 and 120588119886 perp120588119887 which implies that 119875

119899(120588119886) perp 119875

119898(120588119887) for every 119899119898 isin N

witnessing that (119875119899 119875

119898) is orthogonality preserving for every

119899119898 isin NThe proof of (b) follows in a similar manner

We can obtain now a corollary which is a first step towardthe description of orthogonality preserving orthogonallyadditive and holomorphic mappings between Clowast-algebras

Corollary 7 Let 119891 119861119860(0 984858) rarr 119861 be a holomorphic


119896=0119875119896


119860(0 120575) Suppose 119891 is orthogonality preserving and

orthogonally additive on (resp orthogonally additive and zeroproducts preserving)119860

119904119886cap119880 Then there exists a sequence (119879

119899)

of operators from 119860 into 119861 satisfying that the pair (119879119899 119879

119898) is

orthogonality preserving on119860119904119886(resp zero products preserving

on 119860119904119886) for every 119899119898 isin N and

119891 (119909) =

infin

sum

119899=1

119879119899(119909

119899

) (18)

uniformly in 119909 isin 119880 In particular every119879119899is orthogonality pre-

serving (resp zero products preserving) on 119860119904119886 Furthermore

119891 is symmetric if and only if every 119879119899is symmetric

Proof Combining Lemma 3 and Proposition 6 we deducethat 119875

0= 0 119875

119899is orthogonally additive on 119860

119904119886 and (119875

119899 119875

119898)

is orthogonality preserving on 119860119904119886

for every 119899119898 in N ByTheorem 1 for each natural 119899 there exists an operator 119879

119899

119860 rarr 119861 such that 119875119899 le 119879

119899 le 2119875

119899 and

119875119899(119886) = 119879

119899(119886

119899

) (19)

for every 119886 isin 119860Consider now two positive elements 119886 119887 isin 119860 with 119886 perp 119887

and fix 119899119898 isin N In this case there exist positive elements 119888 119889in119860with 119888119899 = 119886 and 119889119898 = 119887 and 119888 perp 119889 Since the pair (119875

119899 119875

119898)

is orthogonality preserving on119860119904119886 we have 119879

119899(119886) = 119879

119899(119888

119899

) =

119875119899(119888) perp 119875

119898(119889) = 119879

119898(119889

119898

) = 119879119898(119887) Now noticing that given

119886 119887 in 119860119904119886with 119886 perp 119887 we can write 119886 = 119886+ minus 119887minus and 119887 = 119887+ minus

119887minus where 119886120590 and 119887120591 are positive 119886+ perp 119886

minus

119887+

perp 119887minus and 119886120590 perp

119887120591 for every 120590 120591 isin + minus we deduce that 119879

119899(119886) perp 119879

119898(119887)

This shows that the pair (119879119899 119879

119898) is orthogonality preserving

on 119860119904119886

When 119891 is orthogonally additive on 119860119904119886and zero prod-

ucts preserving then the pair (119879119899 119879

119898) is zero products

preserving on 119860119904119886

for every 119899119898 isin N The final statementis clear from Lemma 4

It should be remarked here that if a mapping 119891

119861119860(0 120575) rarr 119861 is given by an expression of the form in

(18) which uniformly converges in 119880 = 119861119860(0 120575) where

(119879119899) is a sequence of operators from 119860 into 119861 such that


the pair (119879119899 119879

119898) is orthogonality preserving on 119860

119904119886(resp

zero products preserving on 119860119904119886) for every 119899119898 isin N then

119891 is orthogonally additive and orthogonality preserving on119860

119904119886cap 119880 (resp orthogonally additive on 119860

119904119886cap 119880 and zero

products preserving)

3 Orthogonality Preserving Pairs of Operators

Let 119860 and 119861 be two Clowast-algebras In this section we will studythose pairs of operators 119878 119879 119860 rarr 119861 satisfying that 119878 119879 andthe pair (119878 119879) preserve orthogonality on119860

119904119886 Our description

generalizes some of the results obtained by Wolff in [17]because a (symmetric) mapping 119879 119860 rarr 119861 is orthogonalitypreserving on 119860

119904119886if and only if the pair (119879 119879) enjoys the

same property In particular for every lowast-homomorphismΦ

119860 rarr 119861 the pair (ΦΦ) preserves orthogonality The samestatement is true whenever Φ is a lowast-antihomomorphism ora Jordan lowast-homomorphism or a triple homomorphism forthe triple product 119886 119887 119888 = (12)(119886119887lowast119888 + 119888119887lowast119886)

We observe that 119878 119879 being symmetric implies that (119878 119879)is orthogonality preserving on 119860

119904119886if and only if (119878 119879) is

zero products preserving on 119860119904119886 We shall present here a

newfangled and simplified proof which is also valid for pairsof operators

Let 119886 be an element in a von Neumann algebra 119872 Werecall that the left and right support projections of 119886 (denotedby 119897(119886) and 119889(119886)) are defined as follows 119897(119886) (resp 119889(119886)) isthe smallest projection 119901 isin 119872 (resp 119902 isin 119872) with theproperty that 119901119886 = 119886 (resp 119886119902 = 119886) It is known that when119886 is Hermitian 119889(119886) = 119897(119886) is called the support or rangeprojection of 119886 and is denoted by 119904(119886) It is also known that foreach 119886 = 119886

lowast the sequence (11988613119899

) converges in the strong lowast-topology of119872 to 119904(119886) (cf [18 Sections 110 and 111])

An element 119890 in a Clowast-algebra 119860 is said to be a partialisometry whenever 119890119890lowast119890 = 119890 (equivalently 119890119890lowast or 119890lowast119890 is aprojection in 119860) For each partial isometry 119890 the projections119890119890

lowast and 119890lowast119890 are called the left and right support projectionsassociated with 119890 respectively Every partial isometry 119890 in119860 defines a Jordan product and an involution on 119860

119890(119890) =

119890119890lowast

119860119890lowast

119890 given by 119886∙119890119887 = (12)(119886119890

lowast

119887 + 119887119890lowast

119886) and 119886♯ 119890 = 119890119886lowast

119890

(119886 119887 isin 1198602(119890)) It is known that (119860

2(119890) ∙

119890 ♯

119890) is a unital

JBlowast-algebra with respect to its natural norm and 119890 is the unitelement for the Jordan product ∙

119890

Every element 119886 in a Clowast-algebra 119860 admits a polar decom-position in 119860

lowastlowast that is 119886 decomposes uniquely as follows119886 = 119906|119886| where |119886| = (119886

lowast

119886)12 and 119906 is a partial isometry in

119860lowastlowast such that119906lowast119906 = 119904(|119886|) and119906119906lowast = 119904(|119886lowast|) (cf [18Theorem

1121]) Observe that 119906119906lowast119886 = 119886119906lowast

119906 = 119906 The unique partialisometry 119906 appearing in the polar decomposition of 119886 is calledthe range partial isometry of 119886 and is denoted by 119903(119886) Let usobserve that taking 119888 = 119903(119886)|119886|13 we have 119888119888lowast119888 = 119886 It is alsoeasy to check that for each 119887 isin 119860 with 119887 = 119903(119886)119903(119886)

lowast

119887 (resp119887 = 119887119903(119886)

lowast

119903(119886)) the condition 119886lowast119887 = 0 (resp 119887119886lowast = 0) implies119887 = 0 Furthermore 119886 perp 119887 in 119860 if and only if 119903(119886) perp 119903(119887) in119860lowastlowastWe begin with a basic argument in the study of orthogo-

nality preserving operators betweenClowast-algebras whose proofis inserted here for completeness reasons Let us recall that for

every Clowast-algebra 119860 the multiplier algebra of 119860119872(119860) is theset of all elements 119909 isin 119860

lowastlowast such that for each 119860119909 119909119860 sube 119860We notice that 119872(119860) is a Clowast-algebra and contains the unitelement of 119860lowastlowast

Lemma 8 Let 119860 and 119861 be Clowast-algebras and let 119878 119879 119860 rarr 119861

be a pair of operators(a) The pair (119878 119879) preserves orthogonality (on 119860

119904119886) if

and only if the pair (119878lowastlowast

|119872(119860)

119879lowastlowast

|119872(119860)

) preservesorthogonality (on119872(119860)

119904119886)

(b) The pair (119878 119879) preserves zero products (on 119860119904119886) if

and only if the pair (119878lowastlowast|119872(119860)

119879lowastlowast

|119872(119860)

) preserves zeroproducts (on119872(119860)

119904119886)

Proof (a) The ldquoif rdquo implication is clear Let 119886 119887 be twoelements in 119872(119860) with 119886 perp 119887 We can find two elements119888 and 119889 in 119872(119860) satisfying 119888119888

lowast

119888 = 119886 119889119889lowast119889 = 119887 and119888 perp 119889 Since 119888119909119888 perp 119889119910119889 for every 119909 119910 in 119860 we have119879(119888119909119888) perp 119879(119889119910119889) for every 119909 119910 isin 119860 By Goldstinersquos theoremwe find two bounded nets (119909

120582) and (119910

120583) in 119860 converging in

the weaklowast topology of 119860lowastlowast to 119888lowast and 119889lowast respectively Since119879(119888119909

120582119888)119879(119889119910

120583119889)

lowast

= 119879(119889119910120583119889)

lowast

119879(119888119909120582119888) = 0 for every 120582 120583

119879lowastlowast is weaklowast-continuous the product of 119860lowastlowast is separately

weaklowast-continuous and the involution of 119860lowastlowast is also weaklowast-continuous we get 119879lowastlowast

(119888119888lowast

119888)119879lowastlowast

(119889119889lowast

119889) = 119879lowastlowast

(119886)119879lowastlowast

(119887)lowast

=

0 = 119879lowastlowast

(119887)lowast

119879lowastlowast

(119886) and hence 119879lowastlowast

(119886) perp 119879lowastlowast

(119887) as desiredThe proof of (b) follows by a similar argument

Proposition 9 Let 119878 119879 119860 rarr 119861 be operators between Clowast-algebras such that (119878 119879) is orthogonality preserving on119860

119904119886 Let

us denote ℎ = 119878lowastlowast(1) and 119896 = 119879lowastlowast

(1) Then the identities

119878 (119886) 119879(119886lowast

)

lowast

= 119878 (1198862

) 119896lowast

= ℎ119879((1198862

)

lowast

)

lowast

119879(119886lowast

)

lowast

119878 (119886) = 119896lowast

119878 (1198862

) = 119879((1198862

)

lowast

)

lowast

ℎ

119878 (119886) 119896lowast

= ℎ119879(119886lowast

)

lowast

119896lowast

119878 (119886) = 119879(119886lowast

)

lowast

ℎ

(20)

hold for every 119886 isin 119860

Proof By Lemma 8 we may assume without loss of gener-ality that 119860 is unital (a) for each 120593 isin 119861

lowast the continuousbilinear form 119881

120593 119860 times 119860 rarr C 119881

120593(119886 119887) = 120593(119878(119886)119879(119887

lowast

)lowast

) isorthogonal that is 119881

120593(119886 119887) = 0 whenever 119886119887 = 0 in 119860

119904119886 By

Goldsteinrsquos theorem [19Theorem 110] there exist functionals1205961 120596

2isin 119860

lowast satisfying that

119881120593(119886 119887) = 120596

1(119886119887) + 120596

2(119887119886) (21)

for all 119886 119887 isin 119860 Taking 119887 = 1 and 119886 = 119887 we have

120593 (119878 (119886) 119896lowast

) = 119881120593(119886 1) = 119881

120593(1 119886) = 120593 (ℎ119879(119886)

lowast

)

120593 (119878 (119886) 119879(119886)lowast

) = 120593 (119878 (1198862

) 119896lowast

) = 120593 (ℎ119879(1198862

)

lowast

)

(22)

for every 119886 isin 119860119904119886 respectively Since120593was arbitrarily chosen

we get by linearity 119878(119886)119896lowast = ℎ119879(119886lowast

)lowast and 119878(119886)119879(119886

lowast

)lowast

=

119878(1198862

)119896lowast

= ℎ119879((1198862

)lowast

)lowast for every 119886 isin 119860 The other identities

follow in a similarway but replacing119881120593(119886 119887) = 120593(119878(119886)119879(119887

lowast

)lowast

)

with 119881120593(119886 119887) = 120593(119879(119887

lowast

)lowast

119878(119886))


Lemma 10 Let 1198691 119869

2 119860 rarr 119861 be Jordan lowast-homomorphism

between Clowast-algebras The following statements are equivalent

(a) The pair (1198691 119869


119904119886

(b) The identity

1198691(119886) 119869

2(119886) = 119869

1(119886

2

) 119869lowastlowast

2(1) = 119869

lowastlowast

1(1) 119869

2(119886

2

) (23)

holds for every 119886 isin 119860119904119886

(c) The identity

119869lowastlowast

1(1) 119869

2(119886) = 119869

1(119886) 119869

lowastlowast

2(1) (24)


Furthermore when 119869lowastlowast

1is unital 119869

2(119886) = 119869

1(119886)119869

lowastlowast

2(1) =

119869lowastlowast

2(1)119869

1(119886) for every 119886 in 119860

Proof The implications (a) rArr (b) rArr (c) have beenestablished in Proposition 9 To see (c) rArr (a) we observe that119869119894(119909) = 119869

lowastlowast

119894(1)119869

119894(119909)119869

lowastlowast

119894(1) = 119869

119894(119909)119869

lowastlowast

119894(1) = 119869

lowastlowast

119894(1)119869

119894(119909) for

every 119909 isin 119860 Therefore given 119886 119887 isin 119860119904119886with 119886 perp 119887 we have

1198691(119886)119869

2(119887) = 119869

1(119886)119869

lowastlowast

1(1)119869

2(119887) = 119869

1(119886)119869

1(119887)119869

lowastlowast

2(1) = 0

In [17 Proposition 25] Wolff establishes a uniquenessresult for lowast-homomorphisms between Clowast-algebras showingthat for each pair (119880 119881) of unital lowast-homomorphisms from aunital Clowast-algebra 119860 into a unital Clowast-algebra 119861 the condition(119880 119881) orthogonality preserving on 119860

119904119886implies 119880 = 119881 This

uniqueness result is a direct consequence of our previouslemma

Orthogonality preserving pairs of operators can be alsoused to rediscover the notion of orthomorphism in the senseintroduced by Zaanen in [13] We recall that an operator119879 on a Clowast-algebra 119860 is said to be an orthomorphism or aband preserving operator when the implication 119886 perp 119887 rArr

119879(119886) perp 119887 holds for every 119886 119887 isin 119860 We notice that when 119860is regarded as an 119860-bimodule an operator 119879 119860 rarr 119860 isan orthomorphism if and only if it is a local operator in thesense used by Johnson in [14 Section 3] Clearly an operator119879 119860 rarr 119860 is an orthomorphism if and only if (119879 119868119889

119860)

is orthogonality preserving The following noncommutativeextension of [13 Theorem 5] follows from Proposition 9

Corollary 11 Let 119879 be an operator on a Clowast-algebra119860 Then 119879is an orthomorphism if and only if 119879(119886) = 119879lowastlowast

(1)119886 = 119886119879lowastlowast

(1)for every 119886 in 119860 that is 119879 is a multiple of the identity on 119860 byan element in its center

We recall that two elements 119886 and 119887 in a JBlowast-algebra119860 aresaid to operator commute in 119860 if the multiplication operators119872

119886and119872

119887commute where119872

119886is defined by119872

119886(119909) = 119886∘119909

That is 119886 and 119887 operator commute if and only if (119886 ∘ 119909) ∘ 119887 =119886 ∘ (119909 ∘ 119887) for all 119909 in 119860 A useful result in Jordan theoryassures that self-adjoint elements 119886 and 119887 in 119860 generate aJBlowast-subalgebra that can be realized as a JClowast-subalgebra ofsome 119861(119867) (compare [20]) and under this identification 119886and 119887 commute as elements in 119871(119867) whenever they operatorcommute in 119860 equivalently 1198862 ∘ 119887 = 2(119886 ∘ 119887) ∘ 119886 minus 119886

2

∘ 119887 (cfProposition 1 in [21])

The next lemma contains a property which is probablyknown in Clowast-algebra we include an sketch of the proofbecause we were unable to find an explicit reference

Lemma 12 Let 119890 be a partial isometry in a Clowast-algebra 119860

and let 119886 and 119887 be two elements in 1198602(119890) = 119890119890

lowast

119860119890lowast

119890 Then119886 119887 operator commute in the JBlowast-algebra (119860

2(119890) ∙

119890 ♯

119890) if

and only if 119886119890lowast and 119887119890lowast operator commute in the JBlowast-algebra(119860

2(119890119890

lowast

) ∙119890119890lowast ♯

119890119890lowast) where 119909∙

119890119890lowast119910 = 119909 ∘ 119910 = (12)(119909119910 +

119910119909) for every 119909 119910 isin 1198602(119890119890

lowast

) Furthermore when 119886 and119887 are hermitian elements in (119860

2(119890) ∙

119890 ♯

119890) 119886 and 119887 operator

commute if and only if 119886119890lowast and 119887119890lowast commute in the usual sense(ie 119886119890lowast119887119890lowast = 119887119890lowast119886119890lowast)

Proof We observe that the mapping 119877119890lowast (119860

2(119890) ∙

119890) rarr

(1198602(119890119890

lowast

) ∙119890119890lowast) 119909 997891rarr 119909119890

lowast is a Jordan lowast-isomorphism betweenthe above JBlowast-algebras So the first equivalence is clear Thesecond one has been commented before

Our next corollary relies on the following descriptionof orthogonality preserving operators between Clowast-algebrasobtained in [12] (see also [6])

Theorem 13 (see [12 Theorem 17] [6 Theorem 41 andCorollary 42]) If 119879 is an operator from a Clowast-algebra 119860 intoanother Clowast-algebra 119861 the following are equivalent

(a) 119879 is orthogonality preserving (on 119860119904119886)

(b) There exists a unital Jordan lowast-homomorphism 119869

119872(119860) rarr 119861lowastlowast

2(119903(ℎ)) such that 119869(119909) and ℎ = 119879

lowastlowast

(1)

operator commute and

119879 (119909) = ℎ∙119903(ℎ)

119869 (119909) for every 119909 isin 119860 (25)

where 119872(119860) is the multiplier algebra of 119860 119903(ℎ) isthe range partial isometry of ℎ in 119861

lowastlowast 119861lowastlowast2(119903(ℎ)) =

119903(ℎ)119903(ℎ)lowast

119861lowastlowast

119903(ℎ)lowast

119903(ℎ) and ∙119903(ℎ)

is the natural productmaking 119861lowastlowast

2(119903(ℎ)) a JBlowast-algebra

Furthermore when 119879 is symmetric ℎ is hermitian and hence119903(ℎ) decomposes as orthogonal sum of two projections in 119861lowastlowast

Our next result gives a new perspective for the study oforthogonality preserving (pairs of) operators between Clowast-algebras

Proposition 14 Let 119860 and 119861 be Clowast-algebras Let 119878 119879 119860 rarr

119861 be operators and let ℎ = 119878lowastlowast

(1) and 119896 = 119879lowastlowast

(1) Then thefollowing statements hold

(a) The operator 119878 is orthogonality preserving if and onlyif there exist two Jordan lowast-homomorphisms Φ Φ

119872(119860) rarr 119861lowastlowast satisfying Φ(1) = 119903(ℎ)119903(ℎ)

lowast Φ(1) =119903(ℎ)

lowast

119903(ℎ) and 119878(119886) = Φ(119886)ℎ = ℎΦ(119886) for every 119886 isin 119860(b) 119878 119879 and (119878 119879) are orthogonality preserving on 119860

119904119886if

and only if the following statements hold

(b1) There exist Jordan lowast-homomorphisms Φ1Φ

1

Φ2Φ

2 119872(119860) rarr 119861

lowastlowast satisfying Φ1(1) =

119903(ℎ)119903(ℎ)lowast Φ

1(1) = 119903(ℎ)

lowast

119903(ℎ) Φ2(1) = 119903(119896)


119903(119896)lowastΦ

2(1) = 119903(119896)

lowast

119903(119896)119878(119886) = Φ1(119886)ℎ =

ℎΦ

1(119886) and 119879(119886) = Φ

2(119886)119896 = 119896

Φ

2(119886) for every

119886 isin 119860(b2) The pairs (Φ

1 Φ

2) and (Φ

1Φ

2) are orthogonal-

ity preserving on 119860119904119886

Proof The ldquoif rdquo implications are clear in both statements Wewill only prove the ldquoonly if rdquo implication

(a) By Theorem 13 there exists a unital Jordan lowast-homomorphism 119869

1 119872(119860) rarr 119861

lowastlowast

2(119903(ℎ)) such that

1198691(119909) and ℎ operator commute in the JBlowast-algebra

(119861lowastlowast

2(119903(ℎ)) ∙

119903(ℎ)) and

119878 (119909) = ℎ∙119903(119886)

1198691(119886) for every 119886 isin 119860 (26)

Fix 119886 isin 119860119904119886 Since ℎ and 119869

1(119886) are hermitian

elements in (119861lowastlowast

2(119903(ℎ)) ∙

119903(ℎ)) which operator com-

mute Lemma 12 assures that ℎ119903(ℎ)lowast and 1198691(119886)119903(ℎ)

lowast

commute in the usual sense of 119861lowastlowast that is

ℎ119903(ℎ)lowast

1198691(119886) 119903(ℎ)

lowast

= 1198691(119886) 119903(ℎ)

lowast


(27)

or equivalently


1198691(119886) = 119869

1(119886) 119903(ℎ)

lowast

ℎ (28)

Consequently we have

119878 (119886) = ℎ∙119903(ℎ)

1198691(119886) = ℎ119903(ℎ)

lowast

1198691(119886) = 119869

1(119886) 119903(ℎ)

lowast

ℎ

(29)

for every 119886 isin 119860 The desired statement followsby considering Φ

1(119886) = 119869

1(119886)119903(ℎ)

lowast and Φ

1(119886) =

119903(ℎ)lowast

1198691(119886)

(b) The statement in (b1) follows from (a) We will prove(b2) By hypothesis given 119886 119887 in 119860

119904119886with 119886 perp 119887 we

have

0 = 119878 (119886) 119879(119887)lowast

= (ℎΦ

1(119886)) (119896

Φ

2(119887))

lowast

= ℎΦ

1(119886)

Φ

2(119887)

lowast

119896lowast

(30)

Having in mind that Φ

1(119860) sube 119903(ℎ)

lowast

119903(ℎ)119861lowastlowast

and Φ

2(119860) sube 119861

lowastlowast

119903(119896)lowast

119903(119896) we deduce thatΦ

1(119886)

Φ

2(119887)

lowast

= 0 (compare the comments beforeLemma 8) aswe desired In a similar fashionweproveΦ

2(119887)

lowastΦ

1(119886) = 0 Φ

2(119887)

lowast

Φ1(119886) = 0 = Φ

1(119886)Φ

2(119887)

lowast

4 Holomorphic Mappings Valued ina Commutative Clowast-Algebra

The particular setting in which a holomorphic function isvalued in a commutative Clowast-algebra 119861 provides enoughadvantages to establish a full description of the orthogonallyadditive orthogonality preserving and holomorphic map-pings which are valued in 119861

Proposition 15 Let 119878 119879 119860 rarr 119861 be operators between Clowast-algebras with 119861 commutative Suppose that 119878 119879 and (119878 119879) areorthogonality preserving and let us denote ℎ = 119878

lowastlowast

(1) and119896 = 119879

lowastlowast

(1) Then there exists a Jordan lowast-homomorphismΦ 119872(119860) rarr 119861

lowastlowast satisfyingΦ(1) = 119903(|ℎ|+|119896|) 119878(119886) = Φ(119886)ℎand 119879(119886) = Φ(119886)119896 for every 119886 isin 119860

Proof Let Φ1Φ

1 Φ

2Φ

2 119872(119860) rarr 119861

lowastlowast be the Jordan lowast-homomorphisms satisfying (b1) and (b2) in Proposition 14By hypothesis 119861 is commutative and hence Φ

119894=

Φ

119894for

every 119894 = 1 2 (compare the proof of Proposition 14) Since thepair (Φ

1 Φ


119904119886 Lemma 10

assures that

Φlowastlowast

1(1)Φ

2(119886) = Φ

1(119886)Φ

lowastlowast

2(1) (31)

for every 119886 isin 119860119904119886 In order to simplify notation let us denote

119901 = Φlowastlowast

1(1) and 119902 = Φlowastlowast

2(1)

We define an operator Φ 119872(119860) rarr 119861lowastlowast given by

Φ (119886) = 119901119902Φ1(119886) + 119901 (1 minus 119902)Φ

1(119886) + 119902 (1 minus 119901)Φ

2(119886)

(32)

Since 119901Φ2(119886) = Φ

1(119886)119902 it can be easily checked that Φ is

a Jordan lowast-homomorphism such that 119878(119886) = Φ(119886)ℎ and119879(119886) = Φ(119886)119896 for every 119886 isin 119860

Theorem 16 Let 119891 119861119860(0 984858) rarr 119861 be a holomorphic

mapping where 119860 and 119861 are Clowast-algebras with 119861 commutativeand let 119891 = sum

infin

119896=0119875119896be its Taylor series at zero which

is uniformly converging in 119880 = 119861119860(0 120575) Suppose 119891 is

orthogonality preserving and orthogonally additive on 119860119904119886cap

119880 (equivalently orthogonally additive on 119860119904119886cap 119880 and zero

products preserving) Then there exist a sequence (ℎ119899) in 119861lowastlowast

and a Jordan lowast-homomorphism Φ 119872(119860) rarr 119861lowastlowast such that

119891 (119909) =

infin

sum

119899=1

ℎ119899Φ(119886

119899

) =

infin

sum

119899=1

ℎ119899Φ(119886

119899

) (33)

uniformly in 119886 isin 119880

Proof By Corollary 7 there exists a sequence (119879119899) of oper-

ators from 119860 into 119861 satisfying that the pair (119879119899 119879

119898) is

orthogonality preserving on 119860119904119886(equivalently zero products

preserving on 119860119904119886) for every 119899119898 isin N and

119891 (119909) =

infin

sum

119899=1

119879119899(119909

119899

) (34)

uniformly in 119909 isin 119880 Denote ℎ119899= 119879

lowastlowast

119899(1)

We will prove now the existence of the Jordan lowast-homomorphism Φ We prove by induction that for eachnatural 119899 there exists a Jordan lowast-homomorphism Ψ

119899

119872(119860) rarr 119861lowastlowast such that 119903(Ψ

119899(1)) = 119903(|ℎ

1| + sdot sdot sdot + |ℎ

119899|) and

119879119896(119886) = ℎ

119896Ψ119899(119886) for every 119896 le 119899 119886 isin 119860 The statement for

119899 = 1 follows from Corollary 7 and Proposition 14 Let usassume that our statement is true for 119899 Since for every 119896119898 inN 119879

119896 119879

119898 and the pair (119879

119896 119879

119898) are orthogonality preserving

we can easily check that119879119899+1

1198791+sdot sdot sdot+119879

119899and (119879

119899+1 119879

1+sdot sdot sdot+

119879119899) = (119879

119899+1 (ℎ

1+ sdot sdot sdot + ℎ

119899)Ψ



By Proposition 15 there exists a Jordan lowast-homomorphismΨ119899+1

119872(119860) rarr 119861lowastlowast satisfying 119903(Ψ

119899+1(1)) = 119903(|ℎ

1| + sdot sdot sdot +

|ℎ119899|+|ℎ

119899+1|)119879

119899+1(119886) = ℎ

119899+1Ψ119899+1

(119886119899+1

) and (1198791+sdot sdot sdot+119879

119899)(119886) =

(ℎ1+sdot sdot sdot+ℎ

119899)Ψ

119899+1(119886) for every 119886 isin 119860 Since for each 1 le 119896 le 119899

ℎ119896Ψ119899+1

(119886) = ℎ119896119903 (

1003816100381610038161003816ℎ1

1003816100381610038161003816+ sdot sdot sdot +

1003816100381610038161003816ℎ119899

1003816100381610038161003816+

1003816100381610038161003816ℎ119899+1

1003816100381610038161003816) Ψ

119899+1(119886)

= ℎ119896119903 (1003816100381610038161003816ℎ1

1003816100381610038161003816+ sdot sdot sdot +

1003816100381610038161003816ℎ119899

1003816100381610038161003816) Ψ

119899+1(119886)

= ℎ119896119903 (

1003816100381610038161003816ℎ1

1003816100381610038161003816+ sdot sdot sdot +

1003816100381610038161003816ℎ119899

1003816100381610038161003816) Ψ

119899(119886)=ℎ

119896Ψ119899(119886) = 119879

119896(119886)

(35)

for every 119886 isin 119860 as desiredLet us consider a free ultrafilter U on N By the Banach-

Alaoglu theorem any bounded set in 119861lowastlowast is relatively weaklowast-compact and thus the assignment 119886 997891rarr Φ(119886) = 119908

lowast

minus

limUΨ119899(119886) defines a Jordan lowast-homomorphism from 119872(119860)

into 119861lowastlowast If we fix a natural 119896 we know that 119879119896(119886) = ℎ

119896Ψ119899(119886)

for every 119899 ge 119896 and 119886 isin 119860 Then it can be easily checkedthat 119879

119896(119886) = ℎ

119896Φ(119886) for every 119886 isin 119860 which concludes the

proof

The Banach-Stone type theorem for orthogonally addi-tive orthogonality preserving and holomorphic mappingsbetween commutative Clowast-algebras established inTheorem 2(see [11 Theorem 34]) is a direct consequence of ourprevious result

5 Banach-Stone Type Theorems forHolomorphic Mappings between GeneralClowast-Algebras

In this section we deal with holomorphic functions betweengeneral Clowast-algebras In this more general setting we willrequire additional hypothesis to establish a result in the lineof the aboveTheorem 16

Given a unital Clowast-algebra 119860 the symbol inv(119860) willdenote the set of invertible elements in 119860 The next lemmais a technical tool which is needed later The proof is left tothe reader and follows easily from the fact that inv(119860) is anopen subset of 119860

Lemma 17 Let 119891 119861119860(0 984858) rarr 119861 be a holomorphic

mapping where 119860 and 119861 are Clowast-algebras with 119861 unital andlet 119891 = sum

infin

119896=0119875119896be its Taylor series at zero which is uniformly

converging in 119880 = 119861119860(0 120575) Let us assume that there exists

1198860isin 119880 with 119891(119886

0) isin inv(119861) Then there exists 119898

0isin N such

that sum1198980

119896=0119875119896(119886

0) isin inv(119861)

We can now state a description of those orthogonallyadditive orthogonality preserving and holomorphic map-pings between Clowast-algebras whose image contains an invert-ible element


mapping where 119860 and 119861 are Clowast-algebras with 119861 unital and let119891 = sum

infin

119896=0119875119896be its Taylor series at zero which is uniformly con-

verging in119880 = 119861119860(0 120575) Suppose 119891 is orthogonality preserving

and orthogonally additive on 119860119904119886cap 119880 and 119891(119880) cap inv(119861) = 0




119891 (119886) =

infin

sum

119899=1

ℎ119899

Θ (119886

119899

) =

infin

sum

119899=1

Θ(119886119899

) ℎ119899 (36)


Proof ByCorollary 7 there exists a sequence (119879119899) of operators

from119860 into119861 satisfying that the pair (119879119899 119879

119898) is orthogonality

preserving on 119860119904119886for every 119899119898 isin N and

119891 (119909) =

infin

sum

119899=1

119879119899(119909

119899

) (37)

uniformly in 119909 isin 119880Now Proposition 14 (a) applied to 119879

119899(119899 isin N) implies

the existence of sequences (Φ119899) and (

Φ

119899) of Jordan lowast-

homomorphisms from 119872(119860) into 119861lowastlowast satisfying Φ

119899(1) =

119903(ℎ119899)119903(ℎ

119899)lowast Φ

119899(1) = 119903(ℎ

119899)lowast

119903(ℎ119899) where ℎ

119899= 119879

lowastlowast

119899(1) and

119879119899(119886) = Φ

119899(119886) ℎ

119899= ℎ

119899

Φ

119899(119886) (38)

for every 119886 isin 119860 119899 isin N Moreover from Proposition 14 (b)the pairs (Φ

119899 Φ

119898) and (Φ

119899Φ


on 119860119904119886 for every 119899119898 isin N

Since 119891(119880) cap inv(119861) = 0 it follows from Lemma 17 thatthere exists a natural119898

0and 119886

0isin 119860 such that

1198980

sum

119896=1

119875119896(119886

0) =

1198980

sum

119896=1

Φ119896(119886

119896

0) ℎ

119896=

1198980

sum

119896=1

ℎ119896

Φ

119896(119886

119896

0) isin inv (119861) (39)

We claim that 119903(119903(ℎ1)lowast

119903(ℎ1) + sdot sdot sdot + 119903(ℎ

1198980

)lowast

119903(ℎ1198980

)) = 1

in 119861lowastlowast Otherwise we find a nonzero projection 119902 isin 119861

lowastlowast

satisfying

119903 (119903(ℎ1)

lowast

119903 (ℎ1) + sdot sdot sdot + 119903(ℎ

1198980

)

lowast

119903 (ℎ1198980

)) 119902 = 0 (40)

Since 119903(ℎ119894)lowast

119903(ℎ119894) le 119903(119903(ℎ

1)lowast

119903(ℎ1) + sdot sdot sdot + 119903(ℎ

1198980

)lowast

119903(ℎ1198980

)) thiswould imply that

(

1198980

sum

119896=1

119875119896(119886

0)) 119902 = (

1198980

sum

119896=1

Φ119896(119886

119896

0) ℎ

119896)119902 = 0 (41)

contradicting that sum1198980

119896=1119875119896(119886

0) = sum

1198980

119896=1Φ

119896(119886

119896

0)ℎ

119896is invertible

in 119861Consider now the mapping Ψ = sum

1198980

119896=1

Φ

119896 It is clear that

for each natural 119899 Ψ Φ119899 and the pair (Ψ Φ

119899) are orthog-

onality preserving Applying Proposition 14 (b) we deducethe existence of Jordan lowast-homomorphisms Θ ΘΘ

119899Θ

119899

119872(119860) rarr 119861lowastlowast such that (ΘΘ

119899) and (

Θ

Θ

119899) are orthog-

onality preserving Θ(1) = 119903(119896)119903(119896)lowast Θ(1) = 119903(119896)

lowast

119903(119896)Θ

119899(1) = 119903(ℎ

119899)119903(ℎ

119899)lowast Θ

119899(1) = 119903(ℎ

119899)lowast

119903(ℎ119899)

Ψ (119886) = Θ (119886) 119896 = 119896Θ (119886)

Φ

119899(119886) = Θ

119899(119886) 119903(ℎ

119899)

lowast

119903 (ℎ119899) = 119903(ℎ

119899)

lowast

119903 (ℎ119899)Θ

119899(119886)

(42)


for every 119886 isin 119860 where 119896 = Ψ(1) = 119903(ℎ1)lowast

119903(ℎ1) +

sdot sdot sdot + 119903(ℎ1198980

)lowast

119903(ℎ1198980

) The condition 119903(119896) = 1 proved in theprevious paragraph shows thatΘ(1) = 1 Thus since (Θ Θ

119899)

is orthogonality preserving the last statement in Lemma 10proves that

Θ

119899(119886) =

Θ

119899(1)

Θ (119886) =

Θ (119886)

Θ

119899(1) (43)

for every 119886 isin 119860 119899 isin N The above identities guarantee that

Φ

119899(119886) = Θ (119886) 119903(ℎ

119899)

lowast

119903 (ℎ119899) = 119903(ℎ

119899)

lowast

119903 (ℎ119899)Θ (119886) (44)

for every 119886 isin 119860 119899 isin NA similar argument to the one given above but replac-

ing Φ

119896with Φ

119896 shows the existence of a Jordan lowast-

homomorphism Θ 119872(119860) rarr 119861lowastlowast such that

Φ119899(119886) = Θ (119886) 119903 (ℎ

119899) 119903(ℎ

119899)

lowast

= 119903 (ℎ119899) 119903(ℎ

119899)

lowast

Θ (119886) (45)

for every 119886 isin 119860 119899 isin N which concludes the proof

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are partially supported by the Spanish Min-istry of Economy and Competitiveness DGI Project noMTM2011-23843 and Junta de Andalucıa Grant FQM3737

References

[1] Y Benyamini S Lassalle and J G Llavona ldquoHomogeneousorthogonally additive polynomials on Banach Latticesrdquo Bulletinof the London Mathematical Society vol 38 no 3 pp 459ndash4692006

[2] D Perez-Garcıa and I Villanueva ldquoOrthogonally additivepolynomials on spaces of continuous functionsrdquo Journal ofMathematical Analysis and Applications vol 306 no 1 pp 97ndash105 2005

[3] C Palazuelos A M Peralta and I Villanueva ldquoOrthogonallyadditive polynomials on Clowast-algebrasrdquo Quarterly Journal ofMathematics vol 59 no 3 pp 363ndash374 2008

[4] K Sundaresan ldquoGeometry of spaces of homogeneous polyno-mials on Banach latticesrdquo in Applied Geometry and DiscreteMathematics Discrete Mathematics andTheoretical ComputerScience pp 571ndash586 American Mathematical Society Provi-dence RI USA 1991

[5] D Carando S Lassalle and I Zalduendo ldquoOrthogonallyadditive polynomials over C(K) are measuresmdasha short proofrdquoIntegral Equations and Operator Theory vol 56 no 4 pp 597ndash602 2006

[6] M Burgos F J Fernandez-Polo J J Garces and A M PeraltaldquoOrthogonality preservers revisitedrdquoAsian-European Journal ofMathematics vol 2 no 3 pp 387ndash405 2009

[7] Q Bu and G Buskes ldquoPolynomials on Banach lattices andpositive tensor productsrdquo Journal of Mathematical Analysis andApplications vol 388 no 2 pp 845ndash862 2012

[8] D Carando S Lassalle and I Zalduendo ldquoOrthogonallyadditive holomorphic functions of bounded type over C(K)rdquoProceedings of the Edinburgh Mathematical Society vol 53 no3 pp 609ndash618 2010

[9] J A Jaramillo A Prieto and I Zalduendo ldquoOrthogonallyadditive holomorphic functions on open subsets of C(K)rdquoRevista Matematica Complutense vol 25 no 1 pp 31ndash41 2012

[10] A M Peralta and D Puglisi ldquoOrthogonally additive holomor-phic functions on Clowast-algebrasrdquo Operators and Matrices vol 6no 3 pp 621ndash629 2012

[11] Q Bu M H Hsu and N Ch Wong ldquoZero products and normpreserving orthogonally additive homogeneous polynomials onClowast-algebrasrdquo Preprint

[12] M Burgos F J Fernandez-Polo J J Garces J M Moreno andA M Peralta ldquoOrthogonality preservers in Clowast-algebras JBlowast-algebras and JBlowast-triplesrdquo Journal of Mathematical Analysis andApplications vol 348 no 1 pp 220ndash233 2008

[13] A C Zaanen ldquoExamples of orthomorphismsrdquo Journal ofApproximation Theory vol 13 no 2 pp 192ndash204 1975

[14] B E Johnson ldquoLocal derivations on Clowast-algebras are deriva-tionsrdquo Transactions of the American Mathematical Society vol353 no 1 pp 313ndash325 2001

[15] S Dineen Complex Analysis on Infinite Dimensional SpacesSpringer 1999

[16] T W Gamelin ldquoAnalytic functions on Banach spacesrdquo inComplex Potential Theory vol 439 of NATO Advanced ScienceInstitutes Series C pp 187ndash233 Kluwer Academic DodrechtThe Netherlands 1994

[17] M Wolff ldquoDisjointness preserving operators on Clowast-algebrasrdquoArchiv der Mathematik vol 62 no 3 pp 248ndash253 1994

[18] S Sakai Clowast-Algebras and Wlowast-Algebras Springer Berlin Ger-many 1971

[19] S Goldstein ldquoStationarity of operator algebrasrdquo Journal ofFunctional Analysis vol 118 no 2 pp 275ndash308 1993

[20] J D M Wright ldquoJordan Clowast-algebrasrdquo Michigan MathematicalJournal vol 24 pp 291ndash302 1977

[21] D Topping ldquoJordan algebras of self-adjoint operatorsrdquoMemoirsof the American Mathematical Society vol 53 1965

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


(Φ 1205721 120572

2) where Φ 119862[0 1] rarr C is a lowast-homomorphism

and 1205721 120572

2isin C satisfying that 119875(119891) = 120572

1Φ(119891) + 120572

2Φ(119891

2

) forevery 119891 isin 119862([0 1])

To avoid the difficulties commented above Carandoet al introduce a factorization through an 119871

1(120583) space

More concretely for each compact Hausdorff space 119870 aholomorphic mapping of bounded type 119891 119862(119870) rarr C

is orthogonally additive if and only if there exist a Borelregular measure 120583 on 119870 a sequence (119892

119896)119896sube 119871

1(120583) and a

holomorphic function of bounded type ℎ 119862(119870) rarr 1198711(120583)

such that ℎ(119886) = suminfin

119896=0119892119896119886119896 and

119891 (119886) = int

119870

ℎ (119886) 119889120583 (2)

for every 119886 isin 119862(119870) (cf [8 Theorem 33])When 119862(119870) is replaced with a general Clowast-algebra 119860

a holomorphic function of bounded type 119891 119860 rarr C

is orthogonally additive on 119860119904119886

if and only if there exist apositive functional 120593 in 119860

lowast a sequence (120595119899) in 119871

1(119860

lowastlowast

120593)and a power series holomorphic function ℎ in H

119887(119860 119860

lowast

)

such that

ℎ (119886) =

infin

sum

119896=1

120595119896sdot 119886

119896

119891 (119886) = ⟨1119860lowastlowast ℎ (119886)⟩ = int ℎ (119886) 119889120593

(3)

for every 119886 in 119860 where 1119860lowastlowast denotes the unit element in 119860lowastlowast

and 1198711(119860

lowastlowast

120593) is a noncommutative 1198711-space (cf [10])

A very recent contribution due to Bu et al [11] showsthat for holomorphicmappings between119862(119870) spaces we canavoid the factorization through an 119871

1(120583)-space by imposing

additional hypothesis Before stating the detailed result wewill set down some definitions

Let 119860 and 119861 be Clowast-algebras When 119891 119880 sube 119860 rarr 119861 is amap and the condition

119886 perp 119887 997904rArr 119891 (119886) perp 119891 (119887)

(resp 119886119887 = 0 997904rArr 119891 (119886) 119891 (119887) = 0)

(4)

holds for every 119886 119887 isin 119880 we will say that 119891 preserves orthogo-nality or it is orthogonality preserving (resp 119891 preserves zeroproducts) on 119880 In the case 119860 = 119880 we will simply say that 119891is orthogonality preserving (resp 119891 preserves zero products)Orthogonality preserving bounded linear maps between Clowast-algebras were completely described in [12 Theorem 17] (see[6] for completeness)

The following Banach-Stone type theorem for zero prod-uct preserving or orthogonality preserving holomorphicfunctions between 119862

0(119871) spaces is established by Bu et al in

[11 Theorem 34]

Theorem 2 (see [11]) Let 1198711and 119871

2be locally compact

Hausdorff spaces and let 119891 1198611198620(1198711)(0 119903) rarr 119862

0(119871

2) be a

bounded orthogonally additive holomorphic function If 119891 iszero product preserving or orthogonality preserving then thereexist a sequence (O

119899) of open subsets of 119871

2 a sequence (ℎ

119899)

of bounded functions from 1198712cup infin into C and a mapping

120593 1198712rarr 119871

1such that for each natural 119899 the function ℎ

119899is

continuous and nonvanishing on O119899and

119891 (119886) (119905) =

infin

sum

119899=1

ℎ119899(119905) (119886 (120593 (119905)))

119899

(119905 isin 1198712) (5)

uniformly in 119886 isin 1198611198620(1198711)(0 119903)

The study developed by Bu et al is restricted to com-mutative Clowast-algebras or to orthogonality preserving andorthogonally additive 119899-homogeneous polynomials betweengeneral Clowast-algebras The aim of this paper is to extend theirstudy to holomorphic maps between general Clowast-algebrasIn Section 4 we determine the form of every orthogonalitypreserving and orthogonally additive holomorphic functionfrom a general Clowast-algebra into a commutative Clowast-algebra(see Theorem 16)

In the wider setting of holomorphic mappings betweengeneral Clowast-algebras we prove the following let 119860 and 119861

be Clowast-algebras with 119861 unital and let 119891 119861119860(0 984858) rarr 119861

be a holomorphic mapping whose Taylor series at zero isuniformly converging in some open unit ball 119880 = 119861

119860(0 120575)

Suppose 119891 is orthogonality preserving and orthogonallyadditive on119860

119904119886cap119880 and 119891(119880) contains an invertible element




119891 (119909) =

infin

sum

119899=1

ℎ119899

Θ (119886

119899

) =

infin

sum

119899=1

Θ(119886119899

) ℎ119899 (6)

uniformly in 119886 isin 119880 (see Theorem 18)The main tool to establish our main results is a newfan-

gled investigation on orthogonality preserving pairs of oper-ators between Clowast-algebras developed in Section 3 Amongthe novelties presented in Section 3 we find an innovatingalternative characterization of orthogonality preserving oper-ators between Clowast-algebras which complements the originalone established in [12] (see Proposition 14) Orthogonalitypreserving pairs of operators are also valid to determineorthogonality preserving operators and orthomorphisms orlocal operators on Clowast-algebras in the sense employed byZaanen [13] and Johnson [14] respectively

2 Orthogonally Additive OrthogonalityPreserving and Holomorphic Mappingson Clowast-Algebras

Let 119883 and 119884 be Banach spaces Given a natural 119899 a(continuous) 119899-homogeneous polynomial 119875 from 119883 to 119884 isa mapping 119875 119883 rarr 119884 for which there is a (continuous)119899-linear symmetric operator 119860 119883 times sdot sdot sdot times 119883 rarr 119884 suchthat 119875(119909) = 119860(119909 119909) for every 119909 isin 119883 All polynomialsconsidered in this paper are assumed to be continuous Bya 0-homogeneous polynomial we mean a constant functionThe symbol P(

119899

119883119884) will denote the Banach space of allcontinuous 119899-homogeneous polynomials from 119883 to 119884 withnorm given by 119875 = sup

119909le1119875(119909)

Throughout the paper the word operator will alwaysstand for a bounded linear mapping





0in



119896(119911






infin

sum

119896=0

119875119896(119911

0) (119910 minus 119911

0) (7)



infin



1198751198961119896







120593119875119899(119886) =

1

2120587119894

int

120574

120593119891 (120582119886)

120582119899+1

119889120582 (8)





infin






1198750= 0








119904119886



119896=0119875119896be its



119904119886cap



119896(119860

119904119886) sube 119861

119904119886)





119904119886) if and only if (119879 119878) is





119896=0119875119896




119904119886cap119880) if and only if119875

0= 0

and the pair (119875119899 119875





119899 119875

119898)




119898 le 119862119903


1198750(119905119886) 119875

0(119905119887)

lowast

+ 1198750(119905119886)(

infin

sum

119896=1

119875119896(119905119887))

lowast

+ (

infin

sum

119896=1

119875119896(119905119886))(

infin

sum

119896=0

119875119896(119905119887))

lowast

= 0

(9)

and by homogeneity

1198750(119886) 119875

0(119887)

lowast

= minus 1198750(119886)(

infin

sum

119896=1

119905119896

119875119896(119887))

lowast

+ (

infin

sum

119896=1

119905119896

119875119896(119886))(

infin

sum

119896=0

119905119896

119875119896(119887))

lowast

(10)


0(119887)

lowast




119896) is


lowast


1198751(119905119886) 119875

1(119905119887)

lowast

+ 1198751(119905119886)(

infin

sum

119896=2

119875119896(119905119887))

lowast

+ (

infin

sum

119896=2

119875119896(119905119886))(

infin

sum

119896=1

119875119896(119905119887))

lowast

= 0

(11)


1199052

1198751(119886) 119875

1(119887)

lowast

= minus 1199051198751(119886)(

infin

sum

119896=2

119905119896

119875119896(119887))

lowast

minus (

infin

sum

119896=2

119905119896

119875119896(119886))(

infin

sum

119896=1

119905119896

119875119896(119887))

lowast

(12)


10038171003817100381710038171198751(119886) 119875

1(119887)

lowast1003817100381710038171003817le 119905119862

10038171003817100381710038171198751(119886)

1003817100381710038171003817

infin

sum

119896=2

119887119896

119903119896

119905119896minus2

+ 1199051198622

(

infin

sum

119896=2

119886119896

119903119896

119905119896minus2

)(

infin

sum

119896=1

119887119896

119903119896

119905119896minus1

)

(13)


1(119887)

lowast


119895 119875



1198751(119886) 119875

119899+1(119887)

lowast

+ 119875119899+1

(119886) 1198751(119887)

lowast

= minus1199051198751(119886)(

infin

sum

119895=119899+2

119905119895minus119899minus2

119875119895(119887))

lowast

minus 119905

119899

sum

119896=2

119905119896minus2

119875119896(119886)(

infin

sum

119895=119899+1

119905119895minus119899minus1

119875119895(119887))

lowast

minus 119905119875119899+1

(119886)(

infin

sum

119895=2

119905119895minus2

119875119895(119887))

lowast

minus 119905(

infin

sum

119896=119899+2

119905119896minus119899minus2

119875119896(119886))(

infin

sum

119895=1

119905119895minus1

119875119895(119887))

lowast

(14)


1198751(119886) 119875

119899+1(119887)

lowast

+ 119875119899+1

(119886) 1198751(119887)

lowast

= 0 (15)


1199041198751(119886) 119875

119899+1(119887)

lowast

+ 119904119899+1

119875119899+1

(119886) 1198751(119887)

lowast

= 0 (16)


1198751(119886) 119875

119899+1(119887)

lowast

= 0 (17)


119899+1(119887)

lowast


119896(119887)

lowast

119875119895(119886) = 0 (1 le


119899(119886) perp 119875

119898(119887)


119899(120588119886) perp 119875

119898(120588119887) for every 119899119898 isin N

witnessing that (119875119899 119875






119896=0119875119896





119899)


119898) is



119891 (119909) =

infin

sum

119899=1

119879119899(119909

119899

) (18)





0= 0 119875


119904119886 and (119875

119899 119875

119898)



119899

119860 rarr 119861 such that 119875119899 le 119879

119899 le 2119875

119899 and

119875119899(119886) = 119879

119899(119886

119899

) (19)



119899 119875

119898)


119899(119886) = 119879

119899(119888

119899

) =

119875119899(119888) perp 119875

119898(119889) = 119879

119898(119889

119898




minus

119887+



119899(119886) perp 119879

119898(119887)



on 119860119904119886




preserving on 119860119904119886







the pair (119879119899 119879


119904119886(resp




119904119886cap 119880 and zero










119904119886if and only if (119878 119879) is








119890(119890) =

119890119890lowast

119860119890lowast

119890 given by 119886∙119890119887 = (12)(119886119890

lowast

119887 + 119887119890lowast

119886) and 119886♯ 119890 = 119890119886lowast

119890


2(119890) ∙

119890 ♯

119890) is a unital


119890



lowast





lowast

119887 (resp119887 = 119887119903(119886)

lowast







119904119886) if


|119872(119860)

119879lowastlowast

|119872(119860)


119904119886)



119879lowastlowast

|119872(119860)


119904119886)


lowast


120582) and (119910



120582119888)119879(119889119910

120583119889)

lowast

= 119879(119889119910120583119889)

lowast

119879(119888119909120582119888) = 0 for every 120582 120583



(119888119888lowast


(119889119889lowast



(119887)lowast

=


(119887)lowast

119879lowastlowast





119904119886 Let



119878 (119886) 119879(119886lowast

)

lowast

= 119878 (1198862

) 119896lowast

= ℎ119879((1198862

)

lowast

)

lowast

119879(119886lowast

)

lowast

119878 (119886) = 119896lowast

119878 (1198862

) = 119879((1198862

)

lowast

)

lowast

ℎ

119878 (119886) 119896lowast

= ℎ119879(119886lowast

)

lowast

119896lowast

119878 (119886) = 119879(119886lowast

)

lowast

ℎ

(20)




120593 119860 times 119860 rarr C 119881

120593(119886 119887) = 120593(119878(119886)119879(119887

lowast

)lowast


120593(119886 119887) = 0 whenever 119886119887 = 0 in 119860

119904119886 By


2isin 119860


119881120593(119886 119887) = 120596

1(119886119887) + 120596

2(119887119886) (21)


120593 (119878 (119886) 119896lowast

) = 119881120593(119886 1) = 119881

120593(1 119886) = 120593 (ℎ119879(119886)

lowast

)

120593 (119878 (119886) 119879(119886)lowast

) = 120593 (119878 (1198862

) 119896lowast

) = 120593 (ℎ119879(1198862

)

lowast

)

(22)



)lowast and 119878(119886)119879(119886

lowast

)lowast

=

119878(1198862

)119896lowast

= ℎ119879((1198862

)lowast



lowast

)lowast

)

with 119881120593(119886 119887) = 120593(119879(119887

lowast

)lowast

119878(119886))


Lemma 10 Let 1198691 119869



(a) The pair (1198691 119869


119904119886

(b) The identity

1198691(119886) 119869

2(119886) = 119869

1(119886

2


2(1) = 119869

lowastlowast

1(1) 119869

2(119886

2

) (23)


(c) The identity

119869lowastlowast

1(1) 119869

2(119886) = 119869

1(119886) 119869

lowastlowast

2(1) (24)



1is unital 119869

2(119886) = 119869

1(119886)119869

lowastlowast

2(1) =

119869lowastlowast

2(1)119869

1(119886) for every 119886 in 119860


lowastlowast

119894(1)119869

119894(119909)119869

lowastlowast

119894(1) = 119869

119894(119909)119869

lowastlowast

119894(1) = 119869

lowastlowast

119894(1)119869

119894(119909) for


1198691(119886)119869

2(119887) = 119869

1(119886)119869

lowastlowast

1(1)119869

2(119887) = 119869

1(119886)119869

1(119887)119869

lowastlowast

2(1) = 0


119904119886implies 119880 = 119881 This




119860)



(1)119886 = 119886119879lowastlowast



119886and119872



119886(119909) = 119886∘119909


2





lowast

119860119890lowast


2(119890) ∙

119890 ♯

119890) if


2(119890119890

lowast

) ∙119890119890lowast ♯

119890119890lowast) where 119909∙

119890119890lowast119910 = 119909 ∘ 119910 = (12)(119909119910 +

119910119909) for every 119909 119910 isin 1198602(119890119890

lowast


2(119890) ∙

119890 ♯

119890) 119886 and 119887 operator



2(119890) ∙

119890) rarr

(1198602(119890119890

lowast

) ∙119890119890lowast) 119909 997891rarr 119909119890







2(119903(ℎ)) such that 119869(119909) and ℎ = 119879

lowastlowast

(1)


119879 (119909) = ℎ∙119903(ℎ)

119869 (119909) for every 119909 isin 119860 (25)



119903(ℎ)119903(ℎ)lowast

119861lowastlowast

119903(ℎ)lowast

119903(ℎ) and ∙119903(ℎ)











lowast Φ(1) =119903(ℎ)

lowast


119904119886if



1

Φ2Φ

2 119872(119860) rarr 119861


119903(ℎ)119903(ℎ)lowast Φ

1(1) = 119903(ℎ)

lowast

119903(ℎ) Φ2(1) = 119903(119896)


119903(119896)lowastΦ

2(1) = 119903(119896)

lowast

119903(119896)119878(119886) = Φ1(119886)ℎ =

ℎΦ

1(119886) and 119879(119886) = Φ

2(119886)119896 = 119896

Φ

2(119886) for every


1 Φ

2) and (Φ

1Φ

2) are orthogonal-




1 119872(119860) rarr 119861

lowastlowast



(119861lowastlowast

2(119903(ℎ)) ∙

119903(ℎ)) and

119878 (119909) = ℎ∙119903(119886)

1198691(119886) for every 119886 isin 119860 (26)




2(119903(ℎ)) ∙



lowast



1198691(119886) 119903(ℎ)

lowast

= 1198691(119886) 119903(ℎ)

lowast


(27)

or equivalently


1198691(119886) = 119869

1(119886) 119903(ℎ)

lowast

ℎ (28)


119878 (119886) = ℎ∙119903(ℎ)

1198691(119886) = ℎ119903(ℎ)

lowast

1198691(119886) = 119869

1(119886) 119903(ℎ)

lowast

ℎ

(29)


1(119886) = 119869

1(119886)119903(ℎ)

lowast and Φ

1(119886) =

119903(ℎ)lowast

1198691(119886)


119904119886with 119886 perp 119887 we

have

0 = 119878 (119886) 119879(119887)lowast

= (ℎΦ

1(119886)) (119896

Φ

2(119887))

lowast

= ℎΦ

1(119886)

Φ

2(119887)

lowast

119896lowast

(30)


1(119860) sube 119903(ℎ)

lowast


and Φ

2(119860) sube 119861

lowastlowast

119903(119896)lowast


1(119886)

Φ

2(119887)

lowast


2(119887)

lowastΦ

1(119886) = 0 Φ

2(119887)

lowast

Φ1(119886) = 0 = Φ

1(119886)Φ

2(119887)

lowast




lowastlowast

(1) and119896 = 119879

lowastlowast



Proof Let Φ1Φ

1 Φ

2Φ

2 119872(119860) rarr 119861


119894=

Φ

119894for


1 Φ


119904119886 Lemma 10

assures that

Φlowastlowast

1(1)Φ

2(119886) = Φ

1(119886)Φ

lowastlowast

2(1) (31)




2(1)


Φ (119886) = 119901119902Φ1(119886) + 119901 (1 minus 119902)Φ

1(119886) + 119902 (1 minus 119901)Φ

2(119886)

(32)

Since 119901Φ2(119886) = Φ





infin







119891 (119909) =

infin

sum

119899=1

ℎ119899Φ(119886

119899

) =

infin

sum

119899=1

ℎ119899Φ(119886

119899

) (33)




119898) is



119891 (119909) =

infin

sum

119899=1

119879119899(119909

119899

) (34)


lowastlowast

119899(1)


119899


119899(1)) = 119903(|ℎ


119899|) and

119879119896(119886) = ℎ



119896 119879

119898 and the pair (119879

119896 119879




119899and (119879

119899+1 119879

1+sdot sdot sdot+

119879119899) = (119879

119899+1 (ℎ


119899)Ψ





119899+1(1)) = 119903(|ℎ


|ℎ119899|+|ℎ

119899+1|)119879

119899+1(119886) = ℎ

119899+1Ψ119899+1

(119886119899+1


119899)(119886) =


119899)Ψ


ℎ119896Ψ119899+1

(119886) = ℎ119896119903 (

1003816100381610038161003816ℎ1

1003816100381610038161003816+ sdot sdot sdot +

1003816100381610038161003816ℎ119899

1003816100381610038161003816+

1003816100381610038161003816ℎ119899+1

1003816100381610038161003816) Ψ

119899+1(119886)

= ℎ119896119903 (1003816100381610038161003816ℎ1

1003816100381610038161003816+ sdot sdot sdot +

1003816100381610038161003816ℎ119899

1003816100381610038161003816) Ψ

119899+1(119886)

= ℎ119896119903 (

1003816100381610038161003816ℎ1

1003816100381610038161003816+ sdot sdot sdot +

1003816100381610038161003816ℎ119899

1003816100381610038161003816) Ψ

119899(119886)=ℎ

119896Ψ119899(119886) = 119879

119896(119886)

(35)



lowast

minus



119896Ψ119899(119886)


119896(119886) = ℎ


proof







infin



1198860isin 119880 with 119891(119886


0isin N such

that sum1198980

119896=0119875119896(119886

0) isin inv(119861)




infin







119891 (119886) =

infin

sum

119899=1

ℎ119899

Θ (119886

119899

) =

infin

sum

119899=1

Θ(119886119899

) ℎ119899 (36)






119891 (119909) =

infin

sum

119899=1

119879119899(119909

119899

) (37)




Φ



119899(1) =

119903(ℎ119899)119903(ℎ

119899)lowast Φ

119899(1) = 119903(ℎ

119899)lowast

119903(ℎ119899) where ℎ

119899= 119879

lowastlowast

119899(1) and

119879119899(119886) = Φ

119899(119886) ℎ

119899= ℎ

119899

Φ

119899(119886) (38)


119899 Φ

119898) and (Φ

119899Φ




0and 119886


1198980

sum

119896=1

119875119896(119886

0) =

1198980

sum

119896=1

Φ119896(119886

119896

0) ℎ

119896=

1198980

sum

119896=1

ℎ119896

Φ

119896(119886

119896

0) isin inv (119861) (39)


119903(ℎ1) + sdot sdot sdot + 119903(ℎ

1198980

)lowast

119903(ℎ1198980

)) = 1


lowastlowast

satisfying

119903 (119903(ℎ1)

lowast

119903 (ℎ1) + sdot sdot sdot + 119903(ℎ

1198980

)

lowast

119903 (ℎ1198980

)) 119902 = 0 (40)


119903(ℎ119894) le 119903(119903(ℎ

1)lowast

119903(ℎ1) + sdot sdot sdot + 119903(ℎ

1198980

)lowast

119903(ℎ1198980


(

1198980

sum

119896=1

119875119896(119886

0)) 119902 = (

1198980

sum

119896=1

Φ119896(119886

119896

0) ℎ

119896)119902 = 0 (41)


119896=1119875119896(119886

0) = sum

1198980

119896=1Φ

119896(119886

119896

0)ℎ

119896is invertible


1198980

119896=1

Φ



119899) are orthog-


119899Θ

119899


119899) and (

Θ

Θ

119899) are orthog-


lowast

119903(119896)Θ

119899(1) = 119903(ℎ

119899)119903(ℎ

119899)lowast Θ

119899(1) = 119903(ℎ

119899)lowast

119903(ℎ119899)

Ψ (119886) = Θ (119886) 119896 = 119896Θ (119886)

Φ

119899(119886) = Θ

119899(119886) 119903(ℎ

119899)

lowast

119903 (ℎ119899) = 119903(ℎ

119899)

lowast

119903 (ℎ119899)Θ

119899(119886)

(42)



119903(ℎ1) +


)lowast

119903(ℎ1198980


119899)


Θ

119899(119886) =

Θ

119899(1)

Θ (119886) =

Θ (119886)

Θ

119899(1) (43)


Φ

119899(119886) = Θ (119886) 119903(ℎ

119899)

lowast

119903 (ℎ119899) = 119903(ℎ

119899)

lowast

119903 (ℎ119899)Θ (119886) (44)


ing Φ

119896with Φ



Φ119899(119886) = Θ (119886) 119903 (ℎ

119899) 119903(ℎ

119899)

lowast

= 119903 (ℎ119899) 119903(ℎ

119899)

lowast

Θ (119886) (45)




Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













0in



119896(119911






infin

sum

119896=0

119875119896(119911

0) (119910 minus 119911

0) (7)



infin



1198751198961119896







120593119875119899(119886) =

1

2120587119894

int

120574

120593119891 (120582119886)

120582119899+1

119889120582 (8)





infin






1198750= 0








119904119886



119896=0119875119896be its



119904119886cap



119896(119860

119904119886) sube 119861

119904119886)





119904119886) if and only if (119879 119878) is





119896=0119875119896




119904119886cap119880) if and only if119875

0= 0

and the pair (119875119899 119875





119899 119875

119898)




119898 le 119862119903


1198750(119905119886) 119875

0(119905119887)

lowast

+ 1198750(119905119886)(

infin

sum

119896=1

119875119896(119905119887))

lowast

+ (

infin

sum

119896=1

119875119896(119905119886))(

infin

sum

119896=0

119875119896(119905119887))

lowast

= 0

(9)

and by homogeneity

1198750(119886) 119875

0(119887)

lowast

= minus 1198750(119886)(

infin

sum

119896=1

119905119896

119875119896(119887))

lowast

+ (

infin

sum

119896=1

119905119896

119875119896(119886))(

infin

sum

119896=0

119905119896

119875119896(119887))

lowast

(10)


0(119887)

lowast




119896) is


lowast


1198751(119905119886) 119875

1(119905119887)

lowast

+ 1198751(119905119886)(

infin

sum

119896=2

119875119896(119905119887))

lowast

+ (

infin

sum

119896=2

119875119896(119905119886))(

infin

sum

119896=1

119875119896(119905119887))

lowast

= 0

(11)


1199052

1198751(119886) 119875

1(119887)

lowast

= minus 1199051198751(119886)(

infin

sum

119896=2

119905119896

119875119896(119887))

lowast

minus (

infin

sum

119896=2

119905119896

119875119896(119886))(

infin

sum

119896=1

119905119896

119875119896(119887))

lowast

(12)


10038171003817100381710038171198751(119886) 119875

1(119887)

lowast1003817100381710038171003817le 119905119862

10038171003817100381710038171198751(119886)

1003817100381710038171003817

infin

sum

119896=2

119887119896

119903119896

119905119896minus2

+ 1199051198622

(

infin

sum

119896=2

119886119896

119903119896

119905119896minus2

)(

infin

sum

119896=1

119887119896

119903119896

119905119896minus1

)

(13)


1(119887)

lowast


119895 119875



1198751(119886) 119875

119899+1(119887)

lowast

+ 119875119899+1

(119886) 1198751(119887)

lowast

= minus1199051198751(119886)(

infin

sum

119895=119899+2

119905119895minus119899minus2

119875119895(119887))

lowast

minus 119905

119899

sum

119896=2

119905119896minus2

119875119896(119886)(

infin

sum

119895=119899+1

119905119895minus119899minus1

119875119895(119887))

lowast

minus 119905119875119899+1

(119886)(

infin

sum

119895=2

119905119895minus2

119875119895(119887))

lowast

minus 119905(

infin

sum

119896=119899+2

119905119896minus119899minus2

119875119896(119886))(

infin

sum

119895=1

119905119895minus1

119875119895(119887))

lowast

(14)


1198751(119886) 119875

119899+1(119887)

lowast

+ 119875119899+1

(119886) 1198751(119887)

lowast

= 0 (15)


1199041198751(119886) 119875

119899+1(119887)

lowast

+ 119904119899+1

119875119899+1

(119886) 1198751(119887)

lowast

= 0 (16)


1198751(119886) 119875

119899+1(119887)

lowast

= 0 (17)


119899+1(119887)

lowast


119896(119887)

lowast

119875119895(119886) = 0 (1 le


119899(119886) perp 119875

119898(119887)


119899(120588119886) perp 119875

119898(120588119887) for every 119899119898 isin N

witnessing that (119875119899 119875






119896=0119875119896





119899)


119898) is



119891 (119909) =

infin

sum

119899=1

119879119899(119909

119899

) (18)





0= 0 119875


119904119886 and (119875

119899 119875

119898)



119899

119860 rarr 119861 such that 119875119899 le 119879

119899 le 2119875

119899 and

119875119899(119886) = 119879

119899(119886

119899

) (19)



119899 119875

119898)


119899(119886) = 119879

119899(119888

119899

) =

119875119899(119888) perp 119875

119898(119889) = 119879

119898(119889

119898




minus

119887+



119899(119886) perp 119879

119898(119887)



on 119860119904119886




preserving on 119860119904119886







the pair (119879119899 119879


119904119886(resp




119904119886cap 119880 and zero










119904119886if and only if (119878 119879) is








119890(119890) =

119890119890lowast

119860119890lowast

119890 given by 119886∙119890119887 = (12)(119886119890

lowast

119887 + 119887119890lowast

119886) and 119886♯ 119890 = 119890119886lowast

119890


2(119890) ∙

119890 ♯

119890) is a unital


119890



lowast





lowast

119887 (resp119887 = 119887119903(119886)

lowast







119904119886) if


|119872(119860)

119879lowastlowast

|119872(119860)


119904119886)



119879lowastlowast

|119872(119860)


119904119886)


lowast


120582) and (119910



120582119888)119879(119889119910

120583119889)

lowast

= 119879(119889119910120583119889)

lowast

119879(119888119909120582119888) = 0 for every 120582 120583



(119888119888lowast


(119889119889lowast



(119887)lowast

=


(119887)lowast

119879lowastlowast





119904119886 Let



119878 (119886) 119879(119886lowast

)

lowast

= 119878 (1198862

) 119896lowast

= ℎ119879((1198862

)

lowast

)

lowast

119879(119886lowast

)

lowast

119878 (119886) = 119896lowast

119878 (1198862

) = 119879((1198862

)

lowast

)

lowast

ℎ

119878 (119886) 119896lowast

= ℎ119879(119886lowast

)

lowast

119896lowast

119878 (119886) = 119879(119886lowast

)

lowast

ℎ

(20)




120593 119860 times 119860 rarr C 119881

120593(119886 119887) = 120593(119878(119886)119879(119887

lowast

)lowast


120593(119886 119887) = 0 whenever 119886119887 = 0 in 119860

119904119886 By


2isin 119860


119881120593(119886 119887) = 120596

1(119886119887) + 120596

2(119887119886) (21)


120593 (119878 (119886) 119896lowast

) = 119881120593(119886 1) = 119881

120593(1 119886) = 120593 (ℎ119879(119886)

lowast

)

120593 (119878 (119886) 119879(119886)lowast

) = 120593 (119878 (1198862

) 119896lowast

) = 120593 (ℎ119879(1198862

)

lowast

)

(22)



)lowast and 119878(119886)119879(119886

lowast

)lowast

=

119878(1198862

)119896lowast

= ℎ119879((1198862

)lowast



lowast

)lowast

)

with 119881120593(119886 119887) = 120593(119879(119887

lowast

)lowast

119878(119886))


Lemma 10 Let 1198691 119869



(a) The pair (1198691 119869


119904119886

(b) The identity

1198691(119886) 119869

2(119886) = 119869

1(119886

2


2(1) = 119869

lowastlowast

1(1) 119869

2(119886

2

) (23)


(c) The identity

119869lowastlowast

1(1) 119869

2(119886) = 119869

1(119886) 119869

lowastlowast

2(1) (24)



1is unital 119869

2(119886) = 119869

1(119886)119869

lowastlowast

2(1) =

119869lowastlowast

2(1)119869

1(119886) for every 119886 in 119860


lowastlowast

119894(1)119869

119894(119909)119869

lowastlowast

119894(1) = 119869

119894(119909)119869

lowastlowast

119894(1) = 119869

lowastlowast

119894(1)119869

119894(119909) for


1198691(119886)119869

2(119887) = 119869

1(119886)119869

lowastlowast

1(1)119869

2(119887) = 119869

1(119886)119869

1(119887)119869

lowastlowast

2(1) = 0


119904119886implies 119880 = 119881 This




119860)



(1)119886 = 119886119879lowastlowast



119886and119872



119886(119909) = 119886∘119909


2





lowast

119860119890lowast


2(119890) ∙

119890 ♯

119890) if


2(119890119890

lowast

) ∙119890119890lowast ♯

119890119890lowast) where 119909∙

119890119890lowast119910 = 119909 ∘ 119910 = (12)(119909119910 +

119910119909) for every 119909 119910 isin 1198602(119890119890

lowast


2(119890) ∙

119890 ♯

119890) 119886 and 119887 operator



2(119890) ∙

119890) rarr

(1198602(119890119890

lowast

) ∙119890119890lowast) 119909 997891rarr 119909119890







2(119903(ℎ)) such that 119869(119909) and ℎ = 119879

lowastlowast

(1)


119879 (119909) = ℎ∙119903(ℎ)

119869 (119909) for every 119909 isin 119860 (25)



119903(ℎ)119903(ℎ)lowast

119861lowastlowast

119903(ℎ)lowast

119903(ℎ) and ∙119903(ℎ)











lowast Φ(1) =119903(ℎ)

lowast


119904119886if



1

Φ2Φ

2 119872(119860) rarr 119861


119903(ℎ)119903(ℎ)lowast Φ

1(1) = 119903(ℎ)

lowast

119903(ℎ) Φ2(1) = 119903(119896)


119903(119896)lowastΦ

2(1) = 119903(119896)

lowast

119903(119896)119878(119886) = Φ1(119886)ℎ =

ℎΦ

1(119886) and 119879(119886) = Φ

2(119886)119896 = 119896

Φ

2(119886) for every


1 Φ

2) and (Φ

1Φ

2) are orthogonal-




1 119872(119860) rarr 119861

lowastlowast



(119861lowastlowast

2(119903(ℎ)) ∙

119903(ℎ)) and

119878 (119909) = ℎ∙119903(119886)

1198691(119886) for every 119886 isin 119860 (26)




2(119903(ℎ)) ∙



lowast



1198691(119886) 119903(ℎ)

lowast

= 1198691(119886) 119903(ℎ)

lowast


(27)

or equivalently


1198691(119886) = 119869

1(119886) 119903(ℎ)

lowast

ℎ (28)


119878 (119886) = ℎ∙119903(ℎ)

1198691(119886) = ℎ119903(ℎ)

lowast

1198691(119886) = 119869

1(119886) 119903(ℎ)

lowast

ℎ

(29)


1(119886) = 119869

1(119886)119903(ℎ)

lowast and Φ

1(119886) =

119903(ℎ)lowast

1198691(119886)


119904119886with 119886 perp 119887 we

have

0 = 119878 (119886) 119879(119887)lowast

= (ℎΦ

1(119886)) (119896

Φ

2(119887))

lowast

= ℎΦ

1(119886)

Φ

2(119887)

lowast

119896lowast

(30)


1(119860) sube 119903(ℎ)

lowast


and Φ

2(119860) sube 119861

lowastlowast

119903(119896)lowast


1(119886)

Φ

2(119887)

lowast


2(119887)

lowastΦ

1(119886) = 0 Φ

2(119887)

lowast

Φ1(119886) = 0 = Φ

1(119886)Φ

2(119887)

lowast




lowastlowast

(1) and119896 = 119879

lowastlowast



Proof Let Φ1Φ

1 Φ

2Φ

2 119872(119860) rarr 119861


119894=

Φ

119894for


1 Φ


119904119886 Lemma 10

assures that

Φlowastlowast

1(1)Φ

2(119886) = Φ

1(119886)Φ

lowastlowast

2(1) (31)




2(1)


Φ (119886) = 119901119902Φ1(119886) + 119901 (1 minus 119902)Φ

1(119886) + 119902 (1 minus 119901)Φ

2(119886)

(32)

Since 119901Φ2(119886) = Φ





infin







119891 (119909) =

infin

sum

119899=1

ℎ119899Φ(119886

119899

) =

infin

sum

119899=1

ℎ119899Φ(119886

119899

) (33)




119898) is



119891 (119909) =

infin

sum

119899=1

119879119899(119909

119899

) (34)


lowastlowast

119899(1)


119899


119899(1)) = 119903(|ℎ


119899|) and

119879119896(119886) = ℎ



119896 119879

119898 and the pair (119879

119896 119879




119899and (119879

119899+1 119879

1+sdot sdot sdot+

119879119899) = (119879

119899+1 (ℎ


119899)Ψ





119899+1(1)) = 119903(|ℎ


|ℎ119899|+|ℎ

119899+1|)119879

119899+1(119886) = ℎ

119899+1Ψ119899+1

(119886119899+1


119899)(119886) =


119899)Ψ


ℎ119896Ψ119899+1

(119886) = ℎ119896119903 (

1003816100381610038161003816ℎ1

1003816100381610038161003816+ sdot sdot sdot +

1003816100381610038161003816ℎ119899

1003816100381610038161003816+

1003816100381610038161003816ℎ119899+1

1003816100381610038161003816) Ψ

119899+1(119886)

= ℎ119896119903 (1003816100381610038161003816ℎ1

1003816100381610038161003816+ sdot sdot sdot +

1003816100381610038161003816ℎ119899

1003816100381610038161003816) Ψ

119899+1(119886)

= ℎ119896119903 (

1003816100381610038161003816ℎ1

1003816100381610038161003816+ sdot sdot sdot +

1003816100381610038161003816ℎ119899

1003816100381610038161003816) Ψ

119899(119886)=ℎ

119896Ψ119899(119886) = 119879

119896(119886)

(35)



lowast

minus



119896Ψ119899(119886)


119896(119886) = ℎ


proof







infin



1198860isin 119880 with 119891(119886


0isin N such

that sum1198980

119896=0119875119896(119886

0) isin inv(119861)




infin







119891 (119886) =

infin

sum

119899=1

ℎ119899

Θ (119886

119899

) =

infin

sum

119899=1

Θ(119886119899

) ℎ119899 (36)






119891 (119909) =

infin

sum

119899=1

119879119899(119909

119899

) (37)




Φ



119899(1) =

119903(ℎ119899)119903(ℎ

119899)lowast Φ

119899(1) = 119903(ℎ

119899)lowast

119903(ℎ119899) where ℎ

119899= 119879

lowastlowast

119899(1) and

119879119899(119886) = Φ

119899(119886) ℎ

119899= ℎ

119899

Φ

119899(119886) (38)


119899 Φ

119898) and (Φ

119899Φ




0and 119886


1198980

sum

119896=1

119875119896(119886

0) =

1198980

sum

119896=1

Φ119896(119886

119896

0) ℎ

119896=

1198980

sum

119896=1

ℎ119896

Φ

119896(119886

119896

0) isin inv (119861) (39)


119903(ℎ1) + sdot sdot sdot + 119903(ℎ

1198980

)lowast

119903(ℎ1198980

)) = 1


lowastlowast

satisfying

119903 (119903(ℎ1)

lowast

119903 (ℎ1) + sdot sdot sdot + 119903(ℎ

1198980

)

lowast

119903 (ℎ1198980

)) 119902 = 0 (40)


119903(ℎ119894) le 119903(119903(ℎ

1)lowast

119903(ℎ1) + sdot sdot sdot + 119903(ℎ

1198980

)lowast

119903(ℎ1198980


(

1198980

sum

119896=1

119875119896(119886

0)) 119902 = (

1198980

sum

119896=1

Φ119896(119886

119896

0) ℎ

119896)119902 = 0 (41)


119896=1119875119896(119886

0) = sum

1198980

119896=1Φ

119896(119886

119896

0)ℎ

119896is invertible


1198980

119896=1

Φ



119899) are orthog-


119899Θ

119899


119899) and (

Θ

Θ

119899) are orthog-


lowast

119903(119896)Θ

119899(1) = 119903(ℎ

119899)119903(ℎ

119899)lowast Θ

119899(1) = 119903(ℎ

119899)lowast

119903(ℎ119899)

Ψ (119886) = Θ (119886) 119896 = 119896Θ (119886)

Φ

119899(119886) = Θ

119899(119886) 119903(ℎ

119899)

lowast

119903 (ℎ119899) = 119903(ℎ

119899)

lowast

119903 (ℎ119899)Θ

119899(119886)

(42)



119903(ℎ1) +


)lowast

119903(ℎ1198980


119899)


Θ

119899(119886) =

Θ

119899(1)

Θ (119886) =

Θ (119886)

Θ

119899(1) (43)


Φ

119899(119886) = Θ (119886) 119903(ℎ

119899)

lowast

119903 (ℎ119899) = 119903(ℎ

119899)

lowast

119903 (ℎ119899)Θ (119886) (44)


ing Φ

119896with Φ



Φ119899(119886) = Θ (119886) 119903 (ℎ

119899) 119903(ℎ

119899)

lowast

= 119903 (ℎ119899) 119903(ℎ

119899)

lowast

Θ (119886) (45)




Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119896) is


lowast


1198751(119905119886) 119875

1(119905119887)

lowast

+ 1198751(119905119886)(

infin

sum

119896=2

119875119896(119905119887))

lowast

+ (

infin

sum

119896=2

119875119896(119905119886))(

infin

sum

119896=1

119875119896(119905119887))

lowast

= 0

(11)


1199052

1198751(119886) 119875

1(119887)

lowast

= minus 1199051198751(119886)(

infin

sum

119896=2

119905119896

119875119896(119887))

lowast

minus (

infin

sum

119896=2

119905119896

119875119896(119886))(

infin

sum

119896=1

119905119896

119875119896(119887))

lowast

(12)


10038171003817100381710038171198751(119886) 119875

1(119887)

lowast1003817100381710038171003817le 119905119862

10038171003817100381710038171198751(119886)

1003817100381710038171003817

infin

sum

119896=2

119887119896

119903119896

119905119896minus2

+ 1199051198622

(

infin

sum

119896=2

119886119896

119903119896

119905119896minus2

)(

infin

sum

119896=1

119887119896

119903119896

119905119896minus1

)

(13)


1(119887)

lowast


119895 119875



1198751(119886) 119875

119899+1(119887)

lowast

+ 119875119899+1

(119886) 1198751(119887)

lowast

= minus1199051198751(119886)(

infin

sum

119895=119899+2

119905119895minus119899minus2

119875119895(119887))

lowast

minus 119905

119899

sum

119896=2

119905119896minus2

119875119896(119886)(

infin

sum

119895=119899+1

119905119895minus119899minus1

119875119895(119887))

lowast

minus 119905119875119899+1

(119886)(

infin

sum

119895=2

119905119895minus2

119875119895(119887))

lowast

minus 119905(

infin

sum

119896=119899+2

119905119896minus119899minus2

119875119896(119886))(

infin

sum

119895=1

119905119895minus1

119875119895(119887))

lowast

(14)


1198751(119886) 119875

119899+1(119887)

lowast

+ 119875119899+1

(119886) 1198751(119887)

lowast

= 0 (15)


1199041198751(119886) 119875

119899+1(119887)

lowast

+ 119904119899+1

119875119899+1

(119886) 1198751(119887)

lowast

= 0 (16)


1198751(119886) 119875

119899+1(119887)

lowast

= 0 (17)


119899+1(119887)

lowast


119896(119887)

lowast

119875119895(119886) = 0 (1 le


119899(119886) perp 119875

119898(119887)


119899(120588119886) perp 119875

119898(120588119887) for every 119899119898 isin N

witnessing that (119875119899 119875






119896=0119875119896





119899)


119898) is



119891 (119909) =

infin

sum

119899=1

119879119899(119909

119899

) (18)





0= 0 119875


119904119886 and (119875

119899 119875

119898)



119899

119860 rarr 119861 such that 119875119899 le 119879

119899 le 2119875

119899 and

119875119899(119886) = 119879

119899(119886

119899

) (19)



119899 119875

119898)


119899(119886) = 119879

119899(119888

119899

) =

119875119899(119888) perp 119875

119898(119889) = 119879

119898(119889

119898




minus

119887+



119899(119886) perp 119879

119898(119887)



on 119860119904119886




preserving on 119860119904119886







the pair (119879119899 119879


119904119886(resp




119904119886cap 119880 and zero










119904119886if and only if (119878 119879) is








119890(119890) =

119890119890lowast

119860119890lowast

119890 given by 119886∙119890119887 = (12)(119886119890

lowast

119887 + 119887119890lowast

119886) and 119886♯ 119890 = 119890119886lowast

119890


2(119890) ∙

119890 ♯

119890) is a unital


119890



lowast





lowast

119887 (resp119887 = 119887119903(119886)

lowast







119904119886) if


|119872(119860)

119879lowastlowast

|119872(119860)


119904119886)



119879lowastlowast

|119872(119860)


119904119886)


lowast


120582) and (119910



120582119888)119879(119889119910

120583119889)

lowast

= 119879(119889119910120583119889)

lowast

119879(119888119909120582119888) = 0 for every 120582 120583



(119888119888lowast


(119889119889lowast



(119887)lowast

=


(119887)lowast

119879lowastlowast





119904119886 Let



119878 (119886) 119879(119886lowast

)

lowast

= 119878 (1198862

) 119896lowast

= ℎ119879((1198862

)

lowast

)

lowast

119879(119886lowast

)

lowast

119878 (119886) = 119896lowast

119878 (1198862

) = 119879((1198862

)

lowast

)

lowast

ℎ

119878 (119886) 119896lowast

= ℎ119879(119886lowast

)

lowast

119896lowast

119878 (119886) = 119879(119886lowast

)

lowast

ℎ

(20)




120593 119860 times 119860 rarr C 119881

120593(119886 119887) = 120593(119878(119886)119879(119887

lowast

)lowast


120593(119886 119887) = 0 whenever 119886119887 = 0 in 119860

119904119886 By


2isin 119860


119881120593(119886 119887) = 120596

1(119886119887) + 120596

2(119887119886) (21)


120593 (119878 (119886) 119896lowast

) = 119881120593(119886 1) = 119881

120593(1 119886) = 120593 (ℎ119879(119886)

lowast

)

120593 (119878 (119886) 119879(119886)lowast

) = 120593 (119878 (1198862

) 119896lowast

) = 120593 (ℎ119879(1198862

)

lowast

)

(22)



)lowast and 119878(119886)119879(119886

lowast

)lowast

=

119878(1198862

)119896lowast

= ℎ119879((1198862

)lowast



lowast

)lowast

)

with 119881120593(119886 119887) = 120593(119879(119887

lowast

)lowast

119878(119886))


Lemma 10 Let 1198691 119869



(a) The pair (1198691 119869


119904119886

(b) The identity

1198691(119886) 119869

2(119886) = 119869

1(119886

2


2(1) = 119869

lowastlowast

1(1) 119869

2(119886

2

) (23)


(c) The identity

119869lowastlowast

1(1) 119869

2(119886) = 119869

1(119886) 119869

lowastlowast

2(1) (24)



1is unital 119869

2(119886) = 119869

1(119886)119869

lowastlowast

2(1) =

119869lowastlowast

2(1)119869

1(119886) for every 119886 in 119860


lowastlowast

119894(1)119869

119894(119909)119869

lowastlowast

119894(1) = 119869

119894(119909)119869

lowastlowast

119894(1) = 119869

lowastlowast

119894(1)119869

119894(119909) for


1198691(119886)119869

2(119887) = 119869

1(119886)119869

lowastlowast

1(1)119869

2(119887) = 119869

1(119886)119869

1(119887)119869

lowastlowast

2(1) = 0


119904119886implies 119880 = 119881 This




119860)



(1)119886 = 119886119879lowastlowast



119886and119872



119886(119909) = 119886∘119909


2





lowast

119860119890lowast


2(119890) ∙

119890 ♯

119890) if


2(119890119890

lowast

) ∙119890119890lowast ♯

119890119890lowast) where 119909∙

119890119890lowast119910 = 119909 ∘ 119910 = (12)(119909119910 +

119910119909) for every 119909 119910 isin 1198602(119890119890

lowast


2(119890) ∙

119890 ♯

119890) 119886 and 119887 operator



2(119890) ∙

119890) rarr

(1198602(119890119890

lowast

) ∙119890119890lowast) 119909 997891rarr 119909119890







2(119903(ℎ)) such that 119869(119909) and ℎ = 119879

lowastlowast

(1)


119879 (119909) = ℎ∙119903(ℎ)

119869 (119909) for every 119909 isin 119860 (25)



119903(ℎ)119903(ℎ)lowast

119861lowastlowast

119903(ℎ)lowast

119903(ℎ) and ∙119903(ℎ)











lowast Φ(1) =119903(ℎ)

lowast


119904119886if



1

Φ2Φ

2 119872(119860) rarr 119861


119903(ℎ)119903(ℎ)lowast Φ

1(1) = 119903(ℎ)

lowast

119903(ℎ) Φ2(1) = 119903(119896)


119903(119896)lowastΦ

2(1) = 119903(119896)

lowast

119903(119896)119878(119886) = Φ1(119886)ℎ =

ℎΦ

1(119886) and 119879(119886) = Φ

2(119886)119896 = 119896

Φ

2(119886) for every


1 Φ

2) and (Φ

1Φ

2) are orthogonal-




1 119872(119860) rarr 119861

lowastlowast



(119861lowastlowast

2(119903(ℎ)) ∙

119903(ℎ)) and

119878 (119909) = ℎ∙119903(119886)

1198691(119886) for every 119886 isin 119860 (26)




2(119903(ℎ)) ∙



lowast



1198691(119886) 119903(ℎ)

lowast

= 1198691(119886) 119903(ℎ)

lowast


(27)

or equivalently


1198691(119886) = 119869

1(119886) 119903(ℎ)

lowast

ℎ (28)


119878 (119886) = ℎ∙119903(ℎ)

1198691(119886) = ℎ119903(ℎ)

lowast

1198691(119886) = 119869

1(119886) 119903(ℎ)

lowast

ℎ

(29)


1(119886) = 119869

1(119886)119903(ℎ)

lowast and Φ

1(119886) =

119903(ℎ)lowast

1198691(119886)


119904119886with 119886 perp 119887 we

have

0 = 119878 (119886) 119879(119887)lowast

= (ℎΦ

1(119886)) (119896

Φ

2(119887))

lowast

= ℎΦ

1(119886)

Φ

2(119887)

lowast

119896lowast

(30)


1(119860) sube 119903(ℎ)

lowast


and Φ

2(119860) sube 119861

lowastlowast

119903(119896)lowast


1(119886)

Φ

2(119887)

lowast


2(119887)

lowastΦ

1(119886) = 0 Φ

2(119887)

lowast

Φ1(119886) = 0 = Φ

1(119886)Φ

2(119887)

lowast




lowastlowast

(1) and119896 = 119879

lowastlowast



Proof Let Φ1Φ

1 Φ

2Φ

2 119872(119860) rarr 119861


119894=

Φ

119894for


1 Φ


119904119886 Lemma 10

assures that

Φlowastlowast

1(1)Φ

2(119886) = Φ

1(119886)Φ

lowastlowast

2(1) (31)




2(1)


Φ (119886) = 119901119902Φ1(119886) + 119901 (1 minus 119902)Φ

1(119886) + 119902 (1 minus 119901)Φ

2(119886)

(32)

Since 119901Φ2(119886) = Φ





infin







119891 (119909) =

infin

sum

119899=1

ℎ119899Φ(119886

119899

) =

infin

sum

119899=1

ℎ119899Φ(119886

119899

) (33)




119898) is



119891 (119909) =

infin

sum

119899=1

119879119899(119909

119899

) (34)


lowastlowast

119899(1)


119899


119899(1)) = 119903(|ℎ


119899|) and

119879119896(119886) = ℎ



119896 119879

119898 and the pair (119879

119896 119879




119899and (119879

119899+1 119879

1+sdot sdot sdot+

119879119899) = (119879

119899+1 (ℎ


119899)Ψ





119899+1(1)) = 119903(|ℎ


|ℎ119899|+|ℎ

119899+1|)119879

119899+1(119886) = ℎ

119899+1Ψ119899+1

(119886119899+1


119899)(119886) =


119899)Ψ


ℎ119896Ψ119899+1

(119886) = ℎ119896119903 (

1003816100381610038161003816ℎ1

1003816100381610038161003816+ sdot sdot sdot +

1003816100381610038161003816ℎ119899

1003816100381610038161003816+

1003816100381610038161003816ℎ119899+1

1003816100381610038161003816) Ψ

119899+1(119886)

= ℎ119896119903 (1003816100381610038161003816ℎ1

1003816100381610038161003816+ sdot sdot sdot +

1003816100381610038161003816ℎ119899

1003816100381610038161003816) Ψ

119899+1(119886)

= ℎ119896119903 (

1003816100381610038161003816ℎ1

1003816100381610038161003816+ sdot sdot sdot +

1003816100381610038161003816ℎ119899

1003816100381610038161003816) Ψ

119899(119886)=ℎ

119896Ψ119899(119886) = 119879

119896(119886)

(35)



lowast

minus



119896Ψ119899(119886)


119896(119886) = ℎ


proof







infin



1198860isin 119880 with 119891(119886


0isin N such

that sum1198980

119896=0119875119896(119886

0) isin inv(119861)




infin







119891 (119886) =

infin

sum

119899=1

ℎ119899

Θ (119886

119899

) =

infin

sum

119899=1

Θ(119886119899

) ℎ119899 (36)






119891 (119909) =

infin

sum

119899=1

119879119899(119909

119899

) (37)




Φ



119899(1) =

119903(ℎ119899)119903(ℎ

119899)lowast Φ

119899(1) = 119903(ℎ

119899)lowast

119903(ℎ119899) where ℎ

119899= 119879

lowastlowast

119899(1) and

119879119899(119886) = Φ

119899(119886) ℎ

119899= ℎ

119899

Φ

119899(119886) (38)


119899 Φ

119898) and (Φ

119899Φ




0and 119886


1198980

sum

119896=1

119875119896(119886

0) =

1198980

sum

119896=1

Φ119896(119886

119896

0) ℎ

119896=

1198980

sum

119896=1

ℎ119896

Φ

119896(119886

119896

0) isin inv (119861) (39)


119903(ℎ1) + sdot sdot sdot + 119903(ℎ

1198980

)lowast

119903(ℎ1198980

)) = 1


lowastlowast

satisfying

119903 (119903(ℎ1)

lowast

119903 (ℎ1) + sdot sdot sdot + 119903(ℎ

1198980

)

lowast

119903 (ℎ1198980

)) 119902 = 0 (40)


119903(ℎ119894) le 119903(119903(ℎ

1)lowast

119903(ℎ1) + sdot sdot sdot + 119903(ℎ

1198980

)lowast

119903(ℎ1198980


(

1198980

sum

119896=1

119875119896(119886

0)) 119902 = (

1198980

sum

119896=1

Φ119896(119886

119896

0) ℎ

119896)119902 = 0 (41)


119896=1119875119896(119886

0) = sum

1198980

119896=1Φ

119896(119886

119896

0)ℎ

119896is invertible


1198980

119896=1

Φ



119899) are orthog-


119899Θ

119899


119899) and (

Θ

Θ

119899) are orthog-


lowast

119903(119896)Θ

119899(1) = 119903(ℎ

119899)119903(ℎ

119899)lowast Θ

119899(1) = 119903(ℎ

119899)lowast

119903(ℎ119899)

Ψ (119886) = Θ (119886) 119896 = 119896Θ (119886)

Φ

119899(119886) = Θ

119899(119886) 119903(ℎ

119899)

lowast

119903 (ℎ119899) = 119903(ℎ

119899)

lowast

119903 (ℎ119899)Θ

119899(119886)

(42)



119903(ℎ1) +


)lowast

119903(ℎ1198980


119899)


Θ

119899(119886) =

Θ

119899(1)

Θ (119886) =

Θ (119886)

Θ

119899(1) (43)


Φ

119899(119886) = Θ (119886) 119903(ℎ

119899)

lowast

119903 (ℎ119899) = 119903(ℎ

119899)

lowast

119903 (ℎ119899)Θ (119886) (44)


ing Φ

119896with Φ



Φ119899(119886) = Θ (119886) 119903 (ℎ

119899) 119903(ℎ

119899)

lowast

= 119903 (ℎ119899) 119903(ℎ

119899)

lowast

Θ (119886) (45)




Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










the pair (119879119899 119879


119904119886(resp




119904119886cap 119880 and zero










119904119886if and only if (119878 119879) is








119890(119890) =

119890119890lowast

119860119890lowast

119890 given by 119886∙119890119887 = (12)(119886119890

lowast

119887 + 119887119890lowast

119886) and 119886♯ 119890 = 119890119886lowast

119890


2(119890) ∙

119890 ♯

119890) is a unital


119890



lowast





lowast

119887 (resp119887 = 119887119903(119886)

lowast







119904119886) if


|119872(119860)

119879lowastlowast

|119872(119860)


119904119886)



119879lowastlowast

|119872(119860)


119904119886)


lowast


120582) and (119910



120582119888)119879(119889119910

120583119889)

lowast

= 119879(119889119910120583119889)

lowast

119879(119888119909120582119888) = 0 for every 120582 120583



(119888119888lowast


(119889119889lowast



(119887)lowast

=


(119887)lowast

119879lowastlowast





119904119886 Let



119878 (119886) 119879(119886lowast

)

lowast

= 119878 (1198862

) 119896lowast

= ℎ119879((1198862

)

lowast

)

lowast

119879(119886lowast

)

lowast

119878 (119886) = 119896lowast

119878 (1198862

) = 119879((1198862

)

lowast

)

lowast

ℎ

119878 (119886) 119896lowast

= ℎ119879(119886lowast

)

lowast

119896lowast

119878 (119886) = 119879(119886lowast

)

lowast

ℎ

(20)




120593 119860 times 119860 rarr C 119881

120593(119886 119887) = 120593(119878(119886)119879(119887

lowast

)lowast


120593(119886 119887) = 0 whenever 119886119887 = 0 in 119860

119904119886 By


2isin 119860


119881120593(119886 119887) = 120596

1(119886119887) + 120596

2(119887119886) (21)


120593 (119878 (119886) 119896lowast

) = 119881120593(119886 1) = 119881

120593(1 119886) = 120593 (ℎ119879(119886)

lowast

)

120593 (119878 (119886) 119879(119886)lowast

) = 120593 (119878 (1198862

) 119896lowast

) = 120593 (ℎ119879(1198862

)

lowast

)

(22)



)lowast and 119878(119886)119879(119886

lowast

)lowast

=

119878(1198862

)119896lowast

= ℎ119879((1198862

)lowast



lowast

)lowast

)

with 119881120593(119886 119887) = 120593(119879(119887

lowast

)lowast

119878(119886))


Lemma 10 Let 1198691 119869



(a) The pair (1198691 119869


119904119886

(b) The identity

1198691(119886) 119869

2(119886) = 119869

1(119886

2


2(1) = 119869

lowastlowast

1(1) 119869

2(119886

2

) (23)


(c) The identity

119869lowastlowast

1(1) 119869

2(119886) = 119869

1(119886) 119869

lowastlowast

2(1) (24)



1is unital 119869

2(119886) = 119869

1(119886)119869

lowastlowast

2(1) =

119869lowastlowast

2(1)119869

1(119886) for every 119886 in 119860


lowastlowast

119894(1)119869

119894(119909)119869

lowastlowast

119894(1) = 119869

119894(119909)119869

lowastlowast

119894(1) = 119869

lowastlowast

119894(1)119869

119894(119909) for


1198691(119886)119869

2(119887) = 119869

1(119886)119869

lowastlowast

1(1)119869

2(119887) = 119869

1(119886)119869

1(119887)119869

lowastlowast

2(1) = 0


119904119886implies 119880 = 119881 This




119860)



(1)119886 = 119886119879lowastlowast



119886and119872



119886(119909) = 119886∘119909


2





lowast

119860119890lowast


2(119890) ∙

119890 ♯

119890) if


2(119890119890

lowast

) ∙119890119890lowast ♯

119890119890lowast) where 119909∙

119890119890lowast119910 = 119909 ∘ 119910 = (12)(119909119910 +

119910119909) for every 119909 119910 isin 1198602(119890119890

lowast


2(119890) ∙

119890 ♯

119890) 119886 and 119887 operator



2(119890) ∙

119890) rarr

(1198602(119890119890

lowast

) ∙119890119890lowast) 119909 997891rarr 119909119890







2(119903(ℎ)) such that 119869(119909) and ℎ = 119879

lowastlowast

(1)


119879 (119909) = ℎ∙119903(ℎ)

119869 (119909) for every 119909 isin 119860 (25)



119903(ℎ)119903(ℎ)lowast

119861lowastlowast

119903(ℎ)lowast

119903(ℎ) and ∙119903(ℎ)











lowast Φ(1) =119903(ℎ)

lowast


119904119886if



1

Φ2Φ

2 119872(119860) rarr 119861


119903(ℎ)119903(ℎ)lowast Φ

1(1) = 119903(ℎ)

lowast

119903(ℎ) Φ2(1) = 119903(119896)


119903(119896)lowastΦ

2(1) = 119903(119896)

lowast

119903(119896)119878(119886) = Φ1(119886)ℎ =

ℎΦ

1(119886) and 119879(119886) = Φ

2(119886)119896 = 119896

Φ

2(119886) for every


1 Φ

2) and (Φ

1Φ

2) are orthogonal-




1 119872(119860) rarr 119861

lowastlowast



(119861lowastlowast

2(119903(ℎ)) ∙

119903(ℎ)) and

119878 (119909) = ℎ∙119903(119886)

1198691(119886) for every 119886 isin 119860 (26)




2(119903(ℎ)) ∙



lowast



1198691(119886) 119903(ℎ)

lowast

= 1198691(119886) 119903(ℎ)

lowast


(27)

or equivalently


1198691(119886) = 119869

1(119886) 119903(ℎ)

lowast

ℎ (28)


119878 (119886) = ℎ∙119903(ℎ)

1198691(119886) = ℎ119903(ℎ)

lowast

1198691(119886) = 119869

1(119886) 119903(ℎ)

lowast

ℎ

(29)


1(119886) = 119869

1(119886)119903(ℎ)

lowast and Φ

1(119886) =

119903(ℎ)lowast

1198691(119886)


119904119886with 119886 perp 119887 we

have

0 = 119878 (119886) 119879(119887)lowast

= (ℎΦ

1(119886)) (119896

Φ

2(119887))

lowast

= ℎΦ

1(119886)

Φ

2(119887)

lowast

119896lowast

(30)


1(119860) sube 119903(ℎ)

lowast


and Φ

2(119860) sube 119861

lowastlowast

119903(119896)lowast


1(119886)

Φ

2(119887)

lowast


2(119887)

lowastΦ

1(119886) = 0 Φ

2(119887)

lowast

Φ1(119886) = 0 = Φ

1(119886)Φ

2(119887)

lowast




lowastlowast

(1) and119896 = 119879

lowastlowast



Proof Let Φ1Φ

1 Φ

2Φ

2 119872(119860) rarr 119861


119894=

Φ

119894for


1 Φ


119904119886 Lemma 10

assures that

Φlowastlowast

1(1)Φ

2(119886) = Φ

1(119886)Φ

lowastlowast

2(1) (31)




2(1)


Φ (119886) = 119901119902Φ1(119886) + 119901 (1 minus 119902)Φ

1(119886) + 119902 (1 minus 119901)Φ

2(119886)

(32)

Since 119901Φ2(119886) = Φ





infin







119891 (119909) =

infin

sum

119899=1

ℎ119899Φ(119886

119899

) =

infin

sum

119899=1

ℎ119899Φ(119886

119899

) (33)




119898) is



119891 (119909) =

infin

sum

119899=1

119879119899(119909

119899

) (34)


lowastlowast

119899(1)


119899


119899(1)) = 119903(|ℎ


119899|) and

119879119896(119886) = ℎ



119896 119879

119898 and the pair (119879

119896 119879




119899and (119879

119899+1 119879

1+sdot sdot sdot+

119879119899) = (119879

119899+1 (ℎ


119899)Ψ





119899+1(1)) = 119903(|ℎ


|ℎ119899|+|ℎ

119899+1|)119879

119899+1(119886) = ℎ

119899+1Ψ119899+1

(119886119899+1


119899)(119886) =


119899)Ψ


ℎ119896Ψ119899+1

(119886) = ℎ119896119903 (

1003816100381610038161003816ℎ1

1003816100381610038161003816+ sdot sdot sdot +

1003816100381610038161003816ℎ119899

1003816100381610038161003816+

1003816100381610038161003816ℎ119899+1

1003816100381610038161003816) Ψ

119899+1(119886)

= ℎ119896119903 (1003816100381610038161003816ℎ1

1003816100381610038161003816+ sdot sdot sdot +

1003816100381610038161003816ℎ119899

1003816100381610038161003816) Ψ

119899+1(119886)

= ℎ119896119903 (

1003816100381610038161003816ℎ1

1003816100381610038161003816+ sdot sdot sdot +

1003816100381610038161003816ℎ119899

1003816100381610038161003816) Ψ

119899(119886)=ℎ

119896Ψ119899(119886) = 119879

119896(119886)

(35)



lowast

minus



119896Ψ119899(119886)


119896(119886) = ℎ


proof







infin



1198860isin 119880 with 119891(119886


0isin N such

that sum1198980

119896=0119875119896(119886

0) isin inv(119861)




infin







119891 (119886) =

infin

sum

119899=1

ℎ119899

Θ (119886

119899

) =

infin

sum

119899=1

Θ(119886119899

) ℎ119899 (36)






119891 (119909) =

infin

sum

119899=1

119879119899(119909

119899

) (37)




Φ



119899(1) =

119903(ℎ119899)119903(ℎ

119899)lowast Φ

119899(1) = 119903(ℎ

119899)lowast

119903(ℎ119899) where ℎ

119899= 119879

lowastlowast

119899(1) and

119879119899(119886) = Φ

119899(119886) ℎ

119899= ℎ

119899

Φ

119899(119886) (38)


119899 Φ

119898) and (Φ

119899Φ




0and 119886


1198980

sum

119896=1

119875119896(119886

0) =

1198980

sum

119896=1

Φ119896(119886

119896

0) ℎ

119896=

1198980

sum

119896=1

ℎ119896

Φ

119896(119886

119896

0) isin inv (119861) (39)


119903(ℎ1) + sdot sdot sdot + 119903(ℎ

1198980

)lowast

119903(ℎ1198980

)) = 1


lowastlowast

satisfying

119903 (119903(ℎ1)

lowast

119903 (ℎ1) + sdot sdot sdot + 119903(ℎ

1198980

)

lowast

119903 (ℎ1198980

)) 119902 = 0 (40)


119903(ℎ119894) le 119903(119903(ℎ

1)lowast

119903(ℎ1) + sdot sdot sdot + 119903(ℎ

1198980

)lowast

119903(ℎ1198980


(

1198980

sum

119896=1

119875119896(119886

0)) 119902 = (

1198980

sum

119896=1

Φ119896(119886

119896

0) ℎ

119896)119902 = 0 (41)


119896=1119875119896(119886

0) = sum

1198980

119896=1Φ

119896(119886

119896

0)ℎ

119896is invertible


1198980

119896=1

Φ



119899) are orthog-


119899Θ

119899


119899) and (

Θ

Θ

119899) are orthog-


lowast

119903(119896)Θ

119899(1) = 119903(ℎ

119899)119903(ℎ

119899)lowast Θ

119899(1) = 119903(ℎ

119899)lowast

119903(ℎ119899)

Ψ (119886) = Θ (119886) 119896 = 119896Θ (119886)

Φ

119899(119886) = Θ

119899(119886) 119903(ℎ

119899)

lowast

119903 (ℎ119899) = 119903(ℎ

119899)

lowast

119903 (ℎ119899)Θ

119899(119886)

(42)



119903(ℎ1) +


)lowast

119903(ℎ1198980


119899)


Θ

119899(119886) =

Θ

119899(1)

Θ (119886) =

Θ (119886)

Θ

119899(1) (43)


Φ

119899(119886) = Θ (119886) 119903(ℎ

119899)

lowast

119903 (ℎ119899) = 119903(ℎ

119899)

lowast

119903 (ℎ119899)Θ (119886) (44)


ing Φ

119896with Φ



Φ119899(119886) = Θ (119886) 119903 (ℎ

119899) 119903(ℎ

119899)

lowast

= 119903 (ℎ119899) 119903(ℎ

119899)

lowast

Θ (119886) (45)




Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Lemma 10 Let 1198691 119869



(a) The pair (1198691 119869


119904119886

(b) The identity

1198691(119886) 119869

2(119886) = 119869

1(119886

2


2(1) = 119869

lowastlowast

1(1) 119869

2(119886

2

) (23)


(c) The identity

119869lowastlowast

1(1) 119869

2(119886) = 119869

1(119886) 119869

lowastlowast

2(1) (24)



1is unital 119869

2(119886) = 119869

1(119886)119869

lowastlowast

2(1) =

119869lowastlowast

2(1)119869

1(119886) for every 119886 in 119860


lowastlowast

119894(1)119869

119894(119909)119869

lowastlowast

119894(1) = 119869

119894(119909)119869

lowastlowast

119894(1) = 119869

lowastlowast

119894(1)119869

119894(119909) for


1198691(119886)119869

2(119887) = 119869

1(119886)119869

lowastlowast

1(1)119869

2(119887) = 119869

1(119886)119869

1(119887)119869

lowastlowast

2(1) = 0


119904119886implies 119880 = 119881 This




119860)



(1)119886 = 119886119879lowastlowast



119886and119872



119886(119909) = 119886∘119909


2





lowast

119860119890lowast


2(119890) ∙

119890 ♯

119890) if


2(119890119890

lowast

) ∙119890119890lowast ♯

119890119890lowast) where 119909∙

119890119890lowast119910 = 119909 ∘ 119910 = (12)(119909119910 +

119910119909) for every 119909 119910 isin 1198602(119890119890

lowast


2(119890) ∙

119890 ♯

119890) 119886 and 119887 operator



2(119890) ∙

119890) rarr

(1198602(119890119890

lowast

) ∙119890119890lowast) 119909 997891rarr 119909119890







2(119903(ℎ)) such that 119869(119909) and ℎ = 119879

lowastlowast

(1)


119879 (119909) = ℎ∙119903(ℎ)

119869 (119909) for every 119909 isin 119860 (25)



119903(ℎ)119903(ℎ)lowast

119861lowastlowast

119903(ℎ)lowast

119903(ℎ) and ∙119903(ℎ)











lowast Φ(1) =119903(ℎ)

lowast


119904119886if



1

Φ2Φ

2 119872(119860) rarr 119861


119903(ℎ)119903(ℎ)lowast Φ

1(1) = 119903(ℎ)

lowast

119903(ℎ) Φ2(1) = 119903(119896)


119903(119896)lowastΦ

2(1) = 119903(119896)

lowast

119903(119896)119878(119886) = Φ1(119886)ℎ =

ℎΦ

1(119886) and 119879(119886) = Φ

2(119886)119896 = 119896

Φ

2(119886) for every


1 Φ

2) and (Φ

1Φ

2) are orthogonal-




1 119872(119860) rarr 119861

lowastlowast



(119861lowastlowast

2(119903(ℎ)) ∙

119903(ℎ)) and

119878 (119909) = ℎ∙119903(119886)

1198691(119886) for every 119886 isin 119860 (26)




2(119903(ℎ)) ∙



lowast



1198691(119886) 119903(ℎ)

lowast

= 1198691(119886) 119903(ℎ)

lowast


(27)

or equivalently


1198691(119886) = 119869

1(119886) 119903(ℎ)

lowast

ℎ (28)


119878 (119886) = ℎ∙119903(ℎ)

1198691(119886) = ℎ119903(ℎ)

lowast

1198691(119886) = 119869

1(119886) 119903(ℎ)

lowast

ℎ

(29)


1(119886) = 119869

1(119886)119903(ℎ)

lowast and Φ

1(119886) =

119903(ℎ)lowast

1198691(119886)


119904119886with 119886 perp 119887 we

have

0 = 119878 (119886) 119879(119887)lowast

= (ℎΦ

1(119886)) (119896

Φ

2(119887))

lowast

= ℎΦ

1(119886)

Φ

2(119887)

lowast

119896lowast

(30)


1(119860) sube 119903(ℎ)

lowast


and Φ

2(119860) sube 119861

lowastlowast

119903(119896)lowast


1(119886)

Φ

2(119887)

lowast


2(119887)

lowastΦ

1(119886) = 0 Φ

2(119887)

lowast

Φ1(119886) = 0 = Φ

1(119886)Φ

2(119887)

lowast




lowastlowast

(1) and119896 = 119879

lowastlowast



Proof Let Φ1Φ

1 Φ

2Φ

2 119872(119860) rarr 119861


119894=

Φ

119894for


1 Φ


119904119886 Lemma 10

assures that

Φlowastlowast

1(1)Φ

2(119886) = Φ

1(119886)Φ

lowastlowast

2(1) (31)




2(1)


Φ (119886) = 119901119902Φ1(119886) + 119901 (1 minus 119902)Φ

1(119886) + 119902 (1 minus 119901)Φ

2(119886)

(32)

Since 119901Φ2(119886) = Φ





infin







119891 (119909) =

infin

sum

119899=1

ℎ119899Φ(119886

119899

) =

infin

sum

119899=1

ℎ119899Φ(119886

119899

) (33)




119898) is



119891 (119909) =

infin

sum

119899=1

119879119899(119909

119899

) (34)


lowastlowast

119899(1)


119899


119899(1)) = 119903(|ℎ


119899|) and

119879119896(119886) = ℎ



119896 119879

119898 and the pair (119879

119896 119879




119899and (119879

119899+1 119879

1+sdot sdot sdot+

119879119899) = (119879

119899+1 (ℎ


119899)Ψ





119899+1(1)) = 119903(|ℎ


|ℎ119899|+|ℎ

119899+1|)119879

119899+1(119886) = ℎ

119899+1Ψ119899+1

(119886119899+1


119899)(119886) =


119899)Ψ


ℎ119896Ψ119899+1

(119886) = ℎ119896119903 (

1003816100381610038161003816ℎ1

1003816100381610038161003816+ sdot sdot sdot +

1003816100381610038161003816ℎ119899

1003816100381610038161003816+

1003816100381610038161003816ℎ119899+1

1003816100381610038161003816) Ψ

119899+1(119886)

= ℎ119896119903 (1003816100381610038161003816ℎ1

1003816100381610038161003816+ sdot sdot sdot +

1003816100381610038161003816ℎ119899

1003816100381610038161003816) Ψ

119899+1(119886)

= ℎ119896119903 (

1003816100381610038161003816ℎ1

1003816100381610038161003816+ sdot sdot sdot +

1003816100381610038161003816ℎ119899

1003816100381610038161003816) Ψ

119899(119886)=ℎ

119896Ψ119899(119886) = 119879

119896(119886)

(35)



lowast

minus



119896Ψ119899(119886)


119896(119886) = ℎ


proof







infin



1198860isin 119880 with 119891(119886


0isin N such

that sum1198980

119896=0119875119896(119886

0) isin inv(119861)




infin







119891 (119886) =

infin

sum

119899=1

ℎ119899

Θ (119886

119899

) =

infin

sum

119899=1

Θ(119886119899

) ℎ119899 (36)






119891 (119909) =

infin

sum

119899=1

119879119899(119909

119899

) (37)




Φ



119899(1) =

119903(ℎ119899)119903(ℎ

119899)lowast Φ

119899(1) = 119903(ℎ

119899)lowast

119903(ℎ119899) where ℎ

119899= 119879

lowastlowast

119899(1) and

119879119899(119886) = Φ

119899(119886) ℎ

119899= ℎ

119899

Φ

119899(119886) (38)


119899 Φ

119898) and (Φ

119899Φ




0and 119886


1198980

sum

119896=1

119875119896(119886

0) =

1198980

sum

119896=1

Φ119896(119886

119896

0) ℎ

119896=

1198980

sum

119896=1

ℎ119896

Φ

119896(119886

119896

0) isin inv (119861) (39)


119903(ℎ1) + sdot sdot sdot + 119903(ℎ

1198980

)lowast

119903(ℎ1198980

)) = 1


lowastlowast

satisfying

119903 (119903(ℎ1)

lowast

119903 (ℎ1) + sdot sdot sdot + 119903(ℎ

1198980

)

lowast

119903 (ℎ1198980

)) 119902 = 0 (40)


119903(ℎ119894) le 119903(119903(ℎ

1)lowast

119903(ℎ1) + sdot sdot sdot + 119903(ℎ

1198980

)lowast

119903(ℎ1198980


(

1198980

sum

119896=1

119875119896(119886

0)) 119902 = (

1198980

sum

119896=1

Φ119896(119886

119896

0) ℎ

119896)119902 = 0 (41)


119896=1119875119896(119886

0) = sum

1198980

119896=1Φ

119896(119886

119896

0)ℎ

119896is invertible


1198980

119896=1

Φ



119899) are orthog-


119899Θ

119899


119899) and (

Θ

Θ

119899) are orthog-


lowast

119903(119896)Θ

119899(1) = 119903(ℎ

119899)119903(ℎ

119899)lowast Θ

119899(1) = 119903(ℎ

119899)lowast

119903(ℎ119899)

Ψ (119886) = Θ (119886) 119896 = 119896Θ (119886)

Φ

119899(119886) = Θ

119899(119886) 119903(ℎ

119899)

lowast

119903 (ℎ119899) = 119903(ℎ

119899)

lowast

119903 (ℎ119899)Θ

119899(119886)

(42)



119903(ℎ1) +


)lowast

119903(ℎ1198980


119899)


Θ

119899(119886) =

Θ

119899(1)

Θ (119886) =

Θ (119886)

Θ

119899(1) (43)


Φ

119899(119886) = Θ (119886) 119903(ℎ

119899)

lowast

119903 (ℎ119899) = 119903(ℎ

119899)

lowast

119903 (ℎ119899)Θ (119886) (44)


ing Φ

119896with Φ



Φ119899(119886) = Θ (119886) 119903 (ℎ

119899) 119903(ℎ

119899)

lowast

= 119903 (ℎ119899) 119903(ℎ

119899)

lowast

Θ (119886) (45)




Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119903(119896)lowastΦ

2(1) = 119903(119896)

lowast

119903(119896)119878(119886) = Φ1(119886)ℎ =

ℎΦ

1(119886) and 119879(119886) = Φ

2(119886)119896 = 119896

Φ

2(119886) for every


1 Φ

2) and (Φ

1Φ

2) are orthogonal-




1 119872(119860) rarr 119861

lowastlowast



(119861lowastlowast

2(119903(ℎ)) ∙

119903(ℎ)) and

119878 (119909) = ℎ∙119903(119886)

1198691(119886) for every 119886 isin 119860 (26)




2(119903(ℎ)) ∙



lowast



1198691(119886) 119903(ℎ)

lowast

= 1198691(119886) 119903(ℎ)

lowast


(27)

or equivalently


1198691(119886) = 119869

1(119886) 119903(ℎ)

lowast

ℎ (28)


119878 (119886) = ℎ∙119903(ℎ)

1198691(119886) = ℎ119903(ℎ)

lowast

1198691(119886) = 119869

1(119886) 119903(ℎ)

lowast

ℎ

(29)


1(119886) = 119869

1(119886)119903(ℎ)

lowast and Φ

1(119886) =

119903(ℎ)lowast

1198691(119886)


119904119886with 119886 perp 119887 we

have

0 = 119878 (119886) 119879(119887)lowast

= (ℎΦ

1(119886)) (119896

Φ

2(119887))

lowast

= ℎΦ

1(119886)

Φ

2(119887)

lowast

119896lowast

(30)


1(119860) sube 119903(ℎ)

lowast


and Φ

2(119860) sube 119861

lowastlowast

119903(119896)lowast


1(119886)

Φ

2(119887)

lowast


2(119887)

lowastΦ

1(119886) = 0 Φ

2(119887)

lowast

Φ1(119886) = 0 = Φ

1(119886)Φ

2(119887)

lowast




lowastlowast

(1) and119896 = 119879

lowastlowast



Proof Let Φ1Φ

1 Φ

2Φ

2 119872(119860) rarr 119861


119894=

Φ

119894for


1 Φ


119904119886 Lemma 10

assures that

Φlowastlowast

1(1)Φ

2(119886) = Φ

1(119886)Φ

lowastlowast

2(1) (31)




2(1)


Φ (119886) = 119901119902Φ1(119886) + 119901 (1 minus 119902)Φ

1(119886) + 119902 (1 minus 119901)Φ

2(119886)

(32)

Since 119901Φ2(119886) = Φ





infin







119891 (119909) =

infin

sum

119899=1

ℎ119899Φ(119886

119899

) =

infin

sum

119899=1

ℎ119899Φ(119886

119899

) (33)




119898) is



119891 (119909) =

infin

sum

119899=1

119879119899(119909

119899

) (34)


lowastlowast

119899(1)


119899


119899(1)) = 119903(|ℎ


119899|) and

119879119896(119886) = ℎ



119896 119879

119898 and the pair (119879

119896 119879




119899and (119879

119899+1 119879

1+sdot sdot sdot+

119879119899) = (119879

119899+1 (ℎ


119899)Ψ





119899+1(1)) = 119903(|ℎ


|ℎ119899|+|ℎ

119899+1|)119879

119899+1(119886) = ℎ

119899+1Ψ119899+1

(119886119899+1


119899)(119886) =


119899)Ψ


ℎ119896Ψ119899+1

(119886) = ℎ119896119903 (

1003816100381610038161003816ℎ1

1003816100381610038161003816+ sdot sdot sdot +

1003816100381610038161003816ℎ119899

1003816100381610038161003816+

1003816100381610038161003816ℎ119899+1

1003816100381610038161003816) Ψ

119899+1(119886)

= ℎ119896119903 (1003816100381610038161003816ℎ1

1003816100381610038161003816+ sdot sdot sdot +

1003816100381610038161003816ℎ119899

1003816100381610038161003816) Ψ

119899+1(119886)

= ℎ119896119903 (

1003816100381610038161003816ℎ1

1003816100381610038161003816+ sdot sdot sdot +

1003816100381610038161003816ℎ119899

1003816100381610038161003816) Ψ

119899(119886)=ℎ

119896Ψ119899(119886) = 119879

119896(119886)

(35)



lowast

minus



119896Ψ119899(119886)


119896(119886) = ℎ


proof







infin



1198860isin 119880 with 119891(119886


0isin N such

that sum1198980

119896=0119875119896(119886

0) isin inv(119861)




infin







119891 (119886) =

infin

sum

119899=1

ℎ119899

Θ (119886

119899

) =

infin

sum

119899=1

Θ(119886119899

) ℎ119899 (36)






119891 (119909) =

infin

sum

119899=1

119879119899(119909

119899

) (37)




Φ



119899(1) =

119903(ℎ119899)119903(ℎ

119899)lowast Φ

119899(1) = 119903(ℎ

119899)lowast

119903(ℎ119899) where ℎ

119899= 119879

lowastlowast

119899(1) and

119879119899(119886) = Φ

119899(119886) ℎ

119899= ℎ

119899

Φ

119899(119886) (38)


119899 Φ

119898) and (Φ

119899Φ




0and 119886


1198980

sum

119896=1

119875119896(119886

0) =

1198980

sum

119896=1

Φ119896(119886

119896

0) ℎ

119896=

1198980

sum

119896=1

ℎ119896

Φ

119896(119886

119896

0) isin inv (119861) (39)


119903(ℎ1) + sdot sdot sdot + 119903(ℎ

1198980

)lowast

119903(ℎ1198980

)) = 1


lowastlowast

satisfying

119903 (119903(ℎ1)

lowast

119903 (ℎ1) + sdot sdot sdot + 119903(ℎ

1198980

)

lowast

119903 (ℎ1198980

)) 119902 = 0 (40)


119903(ℎ119894) le 119903(119903(ℎ

1)lowast

119903(ℎ1) + sdot sdot sdot + 119903(ℎ

1198980

)lowast

119903(ℎ1198980


(

1198980

sum

119896=1

119875119896(119886

0)) 119902 = (

1198980

sum

119896=1

Φ119896(119886

119896

0) ℎ

119896)119902 = 0 (41)


119896=1119875119896(119886

0) = sum

1198980

119896=1Φ

119896(119886

119896

0)ℎ

119896is invertible


1198980

119896=1

Φ



119899) are orthog-


119899Θ

119899


119899) and (

Θ

Θ

119899) are orthog-


lowast

119903(119896)Θ

119899(1) = 119903(ℎ

119899)119903(ℎ

119899)lowast Θ

119899(1) = 119903(ℎ

119899)lowast

119903(ℎ119899)

Ψ (119886) = Θ (119886) 119896 = 119896Θ (119886)

Φ

119899(119886) = Θ

119899(119886) 119903(ℎ

119899)

lowast

119903 (ℎ119899) = 119903(ℎ

119899)

lowast

119903 (ℎ119899)Θ

119899(119886)

(42)



119903(ℎ1) +


)lowast

119903(ℎ1198980


119899)


Θ

119899(119886) =

Θ

119899(1)

Θ (119886) =

Θ (119886)

Θ

119899(1) (43)


Φ

119899(119886) = Θ (119886) 119903(ℎ

119899)

lowast

119903 (ℎ119899) = 119903(ℎ

119899)

lowast

119903 (ℎ119899)Θ (119886) (44)


ing Φ

119896with Φ



Φ119899(119886) = Θ (119886) 119903 (ℎ

119899) 119903(ℎ

119899)

lowast

= 119903 (ℎ119899) 119903(ℎ

119899)

lowast

Θ (119886) (45)




Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119899+1(1)) = 119903(|ℎ


|ℎ119899|+|ℎ

119899+1|)119879

119899+1(119886) = ℎ

119899+1Ψ119899+1

(119886119899+1


119899)(119886) =


119899)Ψ


ℎ119896Ψ119899+1

(119886) = ℎ119896119903 (

1003816100381610038161003816ℎ1

1003816100381610038161003816+ sdot sdot sdot +

1003816100381610038161003816ℎ119899

1003816100381610038161003816+

1003816100381610038161003816ℎ119899+1

1003816100381610038161003816) Ψ

119899+1(119886)

= ℎ119896119903 (1003816100381610038161003816ℎ1

1003816100381610038161003816+ sdot sdot sdot +

1003816100381610038161003816ℎ119899

1003816100381610038161003816) Ψ

119899+1(119886)

= ℎ119896119903 (

1003816100381610038161003816ℎ1

1003816100381610038161003816+ sdot sdot sdot +

1003816100381610038161003816ℎ119899

1003816100381610038161003816) Ψ

119899(119886)=ℎ

119896Ψ119899(119886) = 119879

119896(119886)

(35)



lowast

minus



119896Ψ119899(119886)


119896(119886) = ℎ


proof







infin



1198860isin 119880 with 119891(119886


0isin N such

that sum1198980

119896=0119875119896(119886

0) isin inv(119861)




infin







119891 (119886) =

infin

sum

119899=1

ℎ119899

Θ (119886

119899

) =

infin

sum

119899=1

Θ(119886119899

) ℎ119899 (36)






119891 (119909) =

infin

sum

119899=1

119879119899(119909

119899

) (37)




Φ



119899(1) =

119903(ℎ119899)119903(ℎ

119899)lowast Φ

119899(1) = 119903(ℎ

119899)lowast

119903(ℎ119899) where ℎ

119899= 119879

lowastlowast

119899(1) and

119879119899(119886) = Φ

119899(119886) ℎ

119899= ℎ

119899

Φ

119899(119886) (38)


119899 Φ

119898) and (Φ

119899Φ




0and 119886


1198980

sum

119896=1

119875119896(119886

0) =

1198980

sum

119896=1

Φ119896(119886

119896

0) ℎ

119896=

1198980

sum

119896=1

ℎ119896

Φ

119896(119886

119896

0) isin inv (119861) (39)


119903(ℎ1) + sdot sdot sdot + 119903(ℎ

1198980

)lowast

119903(ℎ1198980

)) = 1


lowastlowast

satisfying

119903 (119903(ℎ1)

lowast

119903 (ℎ1) + sdot sdot sdot + 119903(ℎ

1198980

)

lowast

119903 (ℎ1198980

)) 119902 = 0 (40)


119903(ℎ119894) le 119903(119903(ℎ

1)lowast

119903(ℎ1) + sdot sdot sdot + 119903(ℎ

1198980

)lowast

119903(ℎ1198980


(

1198980

sum

119896=1

119875119896(119886

0)) 119902 = (

1198980

sum

119896=1

Φ119896(119886

119896

0) ℎ

119896)119902 = 0 (41)


119896=1119875119896(119886

0) = sum

1198980

119896=1Φ

119896(119886

119896

0)ℎ

119896is invertible


1198980

119896=1

Φ



119899) are orthog-


119899Θ

119899


119899) and (

Θ

Θ

119899) are orthog-


lowast

119903(119896)Θ

119899(1) = 119903(ℎ

119899)119903(ℎ

119899)lowast Θ

119899(1) = 119903(ℎ

119899)lowast

119903(ℎ119899)

Ψ (119886) = Θ (119886) 119896 = 119896Θ (119886)

Φ

119899(119886) = Θ

119899(119886) 119903(ℎ

119899)

lowast

119903 (ℎ119899) = 119903(ℎ

119899)

lowast

119903 (ℎ119899)Θ

119899(119886)

(42)



119903(ℎ1) +


)lowast

119903(ℎ1198980


119899)


Θ

119899(119886) =

Θ

119899(1)

Θ (119886) =

Θ (119886)

Θ

119899(1) (43)


Φ

119899(119886) = Θ (119886) 119903(ℎ

119899)

lowast

119903 (ℎ119899) = 119903(ℎ

119899)

lowast

119903 (ℎ119899)Θ (119886) (44)


ing Φ

119896with Φ



Φ119899(119886) = Θ (119886) 119903 (ℎ

119899) 119903(ℎ

119899)

lowast

= 119903 (ℎ119899) 119903(ℎ

119899)

lowast

Θ (119886) (45)




Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119903(ℎ1) +


)lowast

119903(ℎ1198980


119899)


Θ

119899(119886) =

Θ

119899(1)

Θ (119886) =

Θ (119886)

Θ

119899(1) (43)


Φ

119899(119886) = Θ (119886) 119903(ℎ

119899)

lowast

119903 (ℎ119899) = 119903(ℎ

119899)

lowast

119903 (ℎ119899)Θ (119886) (44)


ing Φ

119896with Φ



Φ119899(119886) = Θ (119886) 119903 (ℎ

119899) 119903(ℎ

119899)

lowast

= 119903 (ℎ119899) 119903(ℎ

119899)

lowast

Θ (119886) (45)




Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article orthogonally additive and...

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