matriceal lebesgue spaces and hölder inequality

JOURNAL OF c© 2005, Scientific Horizon

FUNCTION SPACES AND APPLICATIONS http://www.jfsa.net

Volume 3, Number 3 (2005), 239-249

Matriceal Lebesgue spaces and Holder inequality

Sorina Barza∗ , Dimitri Kravvaritis and Nicolae Popa†

(Communicated by Maria Carro)

2000 Mathematics Subject Classification. 42A16, 42A45, 47B35.

Keywords and phrases. Cesaro sums, Toeplitz matrices, Hilbert modules,

Schur multipliers, Arveson projection.

Abstract. We introduce a class of spaces of infinite matrices similar to the

class of Lebesgue spaces Lp (T) , 1 ≤ p ≤ ∞, and we prove matriceal versions of

Holder inequality.

1. Introduction

Let A = (ai,j)i,j≥1 be an infinite complex matrix. We denote by Ak,

k ∈ Z, the diagonal matrix whose entries a′i,j , satisfy the equation:

a′i,j =

{ai,j if j − i = k,

0 otherwise.(1)

∗The research supported by: KAW 2000.0048 and EUROMMAT-ICA1-CT-2000-7022.

†The research supported by: EURROMMAT-ICA1-CT-2000-7022 and CERES 3-28.

240 Matriceal Lebesgue spaces and Holder inequality

It was remarked first in [1] that there is a similarity between theFourier series

∑k∈Z

akeikt of a function on the torus T = [0, 2π) and the

decomposition∑k∈Z

Ak of an infinite matrix in a sum of diagonal matrices.

Moreover Shields [8] observed the analogy between the convolution oftwo functions and the Schur product A ∗ B = (aij · bij)i,j of the matricesA = (aij)i,j and B = (bij)i,j .

We denote by B(�2) the Banach space of all infinite matrices representingbounded operators acting on �2 with respect to the standard basis, endowedwith the operator norm. We denote also by M(�2) the space of all Schurmultipliers with respect to the norm

‖M‖M(�2) = sup‖A‖B(�2)≤1

‖M ∗ A‖B(�2) .

By reasons determined by Theorem 8.1 in [4] we will also call this spacethe space of matriceal measures.

A Toeplitz matrix M is an infinite matrix with the entries mij = mj−i

for all i, j ≥ 1, where (mj)j is a sequence of complex numbers. Sometimeswe use for this matrix the notation M = (mk)+∞

k=−∞, and we denote by Tthe class of all infinite Toeplitz matrices.

Guided by the results in [4] and [10], which characterize the Toeplitzmatrices in M(�2) and B(�2) by means of the associated Fourier series weextended in [3] some aspects of the classical Fejer’s theory to the case ofinfinite matrices. We introduced there a space of continuous matrices C(�2)and a space of integrable matrices L1(�2) and studied relations betweenthem and the more classical B(�2) and M(�2) and also described some oftheir properties.

An important and well-known fact (see [10]) is the following: B (l2) ∩ Tmay be identified with L∞ (T) in the manner indicated in Theorem 0,and we will write it rather abusive as B (l2) ∩ T = L∞ (T) . Similarly,M (l2)∩T =M (T) (see [4]) with equality of norms. Here of course, M (T)is the space of all regular bounded Borel measures μ on T , endowed withthe norm

‖μ‖ = |μ| (T)

where |μ| is the bounded variation of μ.

So, we identify Toeplitz matrices with functions and we intend to extendsome classical results from the Toeplitz matrices setting to the more generalinfinite matrices.

https://www.researchgate.net/publication/227262991_Some_remarks_on_interpolation_theorems_and_the_boundness_of_the_triangular_projection_in_unitary_matrix_spaces?el=1_x_8&enrichId=rgreq-058b7716-c19b-4f6a-87bf-3c28e61f279d&enrichSource=Y292ZXJQYWdlOzI1ODM3NjgzMjtBUzoxNTI0MjMzMjY4MTgzMTRAMTQxMzM1MTk2MTQ0OA==

https://www.researchgate.net/publication/229713307_A_matriceal_analogue_of_Fejer's_theory?el=1_x_8&enrichId=rgreq-058b7716-c19b-4f6a-87bf-3c28e61f279d&enrichSource=Y292ZXJQYWdlOzI1ODM3NjgzMjtBUzoxNTI0MjMzMjY4MTgzMTRAMTQxMzM1MTk2MTQ0OA==

https://www.researchgate.net/publication/30847031_An_analogue_of_a_Hardy-Littlewood-Fejer_inequality_for_upper_triangular_trace_class_operators?el=1_x_8&enrichId=rgreq-058b7716-c19b-4f6a-87bf-3c28e61f279d&enrichSource=Y292ZXJQYWdlOzI1ODM3NjgzMjtBUzoxNTI0MjMzMjY4MTgzMTRAMTQxMzM1MTk2MTQ0OA==

S. Barza, D. Kravvaritis and N. Popa 241

Starting from these remarks, we consider, in the second section a class ofmatrix spaces similar to Lebesgue spaces Lp (T) , 1 ≤ p ≤ ∞, denoted byLp (l2), such that Lp (l2) ∩ T =Lp (T) , 1 ≤ p ≤ ∞.

We initiate also the study of the matriceal Holder inequality. A naiveextension of the classical Holder inequality is, unfortunately, not valid.

We give some special classes of matrices, called matriceal Hardy spaces,such that this inequality will be true for them.

If A = (aij)i,j≥1 we will denote by At = (aji)i,j≥1 the transpose of A .For every infinite matrix A , let us denote by fA =

∑k∈Z

Akeikt. If

A = (ak)+∞k=−∞ is a Toeplitz matrix, then we get fA =

∑k∈Z

akeikt. Other

notations used in the paper will be introduced as needed.

The following result is well-known: (See [10].)

Theorem 0. A Toeplitz matrix A = (ak)+∞k=−∞ belongs to B(�2) if and

only if the function fA ∈ L∞(T). Moreover

‖A‖B(�2) = ‖fA‖L∞(T).

We recall now that Ak plays the role of the ”kth Fourier coefficient ofthe matrix A.” It is well-known that for each f ∈ L∞(T) whose Fouriercoefficients are an, n ∈ Z, we have

f ∈ C(T) if and only if limn→∞

‖σn(f) − f‖L∞(T) = 0,

where σn(f)(t) =n∑

k=−n

(1 − |k|

n + 1

)akeikt .

So, the following definition (see [3]) is natural:

Definition 1. Let A ∈ B(�2) and σn(A) =n∑

k=−n

Ak

(1 − |k|

n + 1

),

n = 1, 2, . . . , for n ∈ N∗, the matriceal Fejer sum of the order n associatedto A .

Then we call a matrix A to be a continuous matrix and we writeA ∈ C(�2) if the following relation holds:

limn→∞

‖σn(A) − A‖B(�2) = 0 .

Obviously C(�2) endowed with the operator norm becomes a Banachspace.

Theorem 0 allows us to write the formula

[B(�2) ∩ T ]∗ = L∞(T) ,

https://www.researchgate.net/publication/229713307_A_matriceal_analogue_of_Fejer's_theory?el=1_x_8&enrichId=rgreq-058b7716-c19b-4f6a-87bf-3c28e61f279d&enrichSource=Y292ZXJQYWdlOzI1ODM3NjgzMjtBUzoxNTI0MjMzMjY4MTgzMTRAMTQxMzM1MTk2MTQ0OA==


where by [H ]∗ we denote the image of the space H of matrices by thecorrespondence A → fA.

Remark 2. For brevity we write in what follows equations like theprevious one in the following manner:

B(�2) ∩ T = L∞(T) ,

C(�2) ∩ T = C(T) .

In the sequel we use another notation, more appropriate to our aims, forthe entries of the matrix A, namely we put

alk =

{al,l+k k ≥ 0, l = 1, 2, 3 . . .

al−k,l k < 0, l = 1, 2, 3, . . .,(2)

and denote A sometimes as A = (alk)k∈Z, l≥1.

Let A(l) = (bmk )k∈Z, m≥1, where l ∈ N∗, be the matrix given by

bmk =

{al

k if m = l

0 if m = l .(3)

We call the matrix A(l), the lth -corner matrix associated to A .Now, if for any corner-matrix A(l) = (bm

k )k∈Z, m≥1 we associate adistribution on T, denoted by fl such that bl

k = fl(k), we get, in caseA ∈ B(�2) ∩ T , that fl = f ∈ L∞(T), for all l ∈ N∗.

We can identify the matrix A = (A(l))l∈N∗ with its sequence of associateddistributions f not= (fl)l∈N∗ , writing this fact as

A = Af .

By Theorem 0 we have the following correspondences:

f ∈ L∞(T) if and only if Af ∈ B(�2) ∩ T , where f = (f, f, f, . . . )

g ∈ L∞(T) if and only if Ag ∈ B(�2) ∩ T , where g = (g, g, g, . . . ) .

Then, of course, it follows

fg ∈ L∞(T) if and only if Afg ∈ B(�2) ∩ T , where fg = (fg, fg, fg, . . . ) .

The matrix A = (aij) is said to be of n-band type if aij = 0 for|i − j| > n.

Having these notions in mind we introduce a commutative product ofinfinite matrices:


Definition 3. Let A = Af and B = Ag two infinite matrices of finiteband type. We introduce now the commutative product � given by

A�Bdef= Afg.

Remark 4. (1) We mention that in the previous definition we tookA = Af , B = Ag infinite matrices of finite band type since, f and g

being trigonometric polynomials, we may consider the product fg.

(2) This product can be defined also for all matrices A, B ∈ B(�2), butA�B does not belong in general to B(�2). Take, for instance, A = Af , withf = (f, 0, 0, . . . ), and B = Ag, with g = (g, 0, 0, . . . ), where f, g ∈ H2(T)such that fg ∈ H1(T) \ H2(T).

(3) Of course, if Af , Ag ∈ B(�2)∩T then it follows that Af�Ag = Afg ∈B(�2) ∩ T .

2. Lebesgue spaces of infinite matrices andHolder inequality

We intend now to introduce some spaces of infinite matrices which aresimilar to classical Lebesgue spaces Lp(T), 1 ≤ p ≤ ∞.

We recall here the following: (See [3]. )

Definition 5. Let L1 (�2) = {A ∈ M(�2) such that limn→∞

‖σn (A) − A‖M(�2)

= 0} . We consider on L1(�2) the induced norm of M(�2).

We need also a space of infinite matrices which is similar to Hilbert spaceL2 (T) and, in fact, extends it.

Let first consider the space

L2 (�2)def=

{A infinite matrices such that PT A ∈ B (�2, �∞) and

[(I − PT ) (A)]t ∈ B (�2, �∞)

},

where I is the identity operator,

PT (A) =

{ai,j if i ≤ j,

0 otherwise,

and B (�2, �∞) is the space of all matrices representing bounded linearoperators from �2 into �∞, endowed with operator norm.

On L2 (�2) we introduce the norm ‖A‖L2(�2)= sup

l∈Z

(∞∑

j=1

∣∣∣ajl

∣∣∣2) 12

.


Motivated by Definitions 1 and 5 we consider the space

L2(�2)def={A∈L2 (�2) such that lim

n→∞σn(A) = A in the norm of L2 (�2)

}.

It is easy to see thatL2 (�2) ⊂ L1 (�2)

and also L2 (�2) ∩ T = L2 (T) .

L2(�2) can be regarded as a matriceal extension of the familiar Hilbertspace L2(T).

A good reason for this is the fact that we may introduce in L2(�2) anextension of the usual scalar product.

We consider now the Arveson projection (see [2]) P : B (�2) → B (�2)∩Tgiven by a fixed Banach limit Λ. More precisely, for A = (al

j)j∈Z, l≥1 andaj = (al

j)l≥1, we put P (A) = (Λ (aj))+∞j=−∞ .

It is easy to see, using the definitions, that:

Theorem 6. The Arveson projection associated to some Banach limitΛ , P (A) =

∑j∈Z

Λ (aj) , where A = (alj)j∈Z, l≥1, maps continuously C (�2)

onto C (�2) ∩ T , and moreover, ‖P‖ ≤ ‖Λ‖.

Now we give the analogue of Theorem 6 for L1 (�2) .

Theorem 7. Let P be the Arveson projection. Then P : M (�2) →M (�2)∩ T and P : L1 (�2) → L1 (�2)∩T are continuous linear maps with‖P‖ ≤ ‖Λ‖.

Proof. Let A = (alj)j∈Z, l≥1. We have:

‖P (A)‖M(�2)=

∥∥∥∥∥∞∑

k=−∞Λ (ak) eikt

∥∥∥∥∥M(T)

= (by Theorem 16.3.5, [7])

= sup‖g‖L∞≤1

∥∥∥∥∥+∞∑

k=−∞Λ(aj)eikt ∗ g

∥∥∥∥∥L∞(T)

= sup‖B‖≤1, B∈T

‖P (A) ∗ B‖B(�2)

= sup‖B‖≤1,B∈T

‖P (A ∗ B)‖B(�2)

≤ (by [2]) ≤ ‖Λ‖ · sup‖B‖≤1

‖A ∗ B‖B(�2)

= ‖Λ‖ · ‖A‖M(�2) .


Hence P : M (�2) → M (T) is a bounded projection with ‖P‖ ≤ ‖Λ‖ .

By Definition 5 this proves also the assertion concerning L1(�2). �

Obviously C (�2) � B(�2) � L2(�2) � L1 (�2) .

It is also easy to see that:

Theorem 8. The Arveson projection P, given by a fixed Banach limitΛ, maps continuously L2(�2) onto L2(�2) ∩ T and ‖P‖ ≤ ‖Λ‖.

So, by Theorem 6 , it follows that C(�2) ∩ T = C(T) is a complementedsubspace of C(�2) with respect to the Arveson projection P and byTheorem 8 the space L2(�2) ∩ T = L2(T) is a complemented subspaceof L2(�2).

We may consider now the complex interpolation space (see [6], [9])[C (�2) , L2 (�2)

]2p

and call it the matriceal Lebesgue space Lp(�2), for2 ≤ p ≤ ∞.

Now we can use a general result of interpolation theory (See [9], 1.17.1,Theorem 1.)

Theorem A. Let (A0, A1) be an interpolation pair of Banach spacesand B a complemented subspace of A0 + A1, such that the correspondingprojection P maps continuously A0 into A0 and A1 into A1 . Let F be aninterpolation functor. Then (A0 ∩ B, A1 ∩ B) is an interpolation pair and

F (A0 ∩ B, A1 ∩ B) = F (A0, A1) ∩ B.

Indeed we put in Theorem A A0 = B(�2), A1 = L2(�2), B =L2(�2) ∩ T = L2(T), P : L2(�2) → L2(T) the Arveson projection andF = [C(�2), L2(�2)] 2

p, for 2 ≤ p ≤ ∞. Then, by Theorems 6 and 8, B

and P satisfy the conditions of Theorem A, hence (C(T), L2(T)) is aninterpolation pair and

Lp(T) = [C(T), L2(T)] 2p

= [C(�2), L2(�2)] 2p∩ T = Lp(�2) ∩ T .

Similarly, using Theorems 7 and 8 in Theorem A, we get

Lp(T) = [L2(T), L1(T)] 2−pp

= [L2(�2), L1(�2)] 2−pp

∩ T = Lp(�2) ∩ T ,

for 1 ≤ p ≤ 2.

Here, of course, we denote [L2(�2), L1(�2)] 2−pp

by Lp(�2), for 1 ≤ p ≤ 2.

In order to justify the introduction of above matrix spaces we would like toprove a version of Holder inequality in a subspace of Lp(�2), for 2 ≤ p ≤ ∞.


Definition 9. We denote by H1(�2) = {A ∈ L1(�2); A upper triangular}and we call H1(�2) the matriceal Hardy space of order 1.

Then H1(�2) is a Banach subspace of L1(�2) and A ∈ H1(�2)∩T if andonly if fA ∈ H1(T).

Similarly H∞(�2) = {A ∈ B(�2); A upper triangular }. Of courseH∞(�2) is a Banach subspace of B(�2) and A ∈ H∞(�2)∩ T if and only iffA ∈ H∞(T).

In what follows we consider only upper triangular matrices.

Then, by straightforward computation, we get:

Remark 10. Let A ∈ T such that fA(t) be an analytic polynomialPn(eit) and B an upper triangular matrix. Then

A�B = B · A,

where A · B means the usual non commutative product of matrices.

Now we are interested to find a matriceal version of the well-known Holderinequality.

At the first glance we are tempted to look at the following naıve matricealHolder inequality in case p = 1, q = ∞.

A ∈ B(�2) and B ∈ L1(�2) implies A�B ∈ L1(�2).

Unfortunately the previous statement does not hold. For instance:

Example 11. Let A = (f, 0, 0, . . . ) ∈ B(�2), where f(t) = 1 +∞∑

k=1

1k

cos kt

and let B = (g, 0, 0, . . . ) ∈ L1(�2), for g(t) =∞∑

k=1

1ln k

eikt.

Then it follows easily that (A�B)10 =∞∑

k=2

1k ln k

= ∞.

However, clearly if A ∈ B(�2) ∩ T and if B ∈ L1(�2) ∩ T then it followsthat A�B ∈ L1(�2) ∩ T .

Of course there are also other classes of matrices A, B such thatA�B ∈ L1(�2).

Theorem 12. Let A and B upper triangular matrices such that A ∈ Twith fA ∈ A(D), and B ∈ H1(�2). Then A�B ∈ H1(�2) and

‖A�B‖H1(�2) ≤ ‖B‖H1(�2) · ‖fA‖A(D).


Proof. It is enough to prove theorem in the case fA equal to the analytic

polynomial Pn(t) =n∑

k=0

akeikt.

Then, for any C ∈ B(�2), we get, using Remark 10,

(A�B) ∗ C = (B · A) ∗ C = B ∗ (a0C + a1C1 + · · · + anCn),

where

C1 =

⎛⎜⎝ 0 c11 . . . c1n . . .0 c21 . . . c2n . . ....

......

......

⎞⎟⎠ , . . . ,

Cn =

⎛⎜⎝ 0 . . . 0 c11 c12 . . .0 . . . 0 c21 c22 . . ....

......

......

...

⎞⎟⎠ .

Thus

‖(A�B) ∗ C‖B(�2) = ‖B ∗ (a0C + a1C1 + · · · + anCn)‖B(�2) ≤ ‖B‖H1(�2)·

‖a0C + a1C1 + · · · + anCn‖B(�2).

But, defining by τ−k : �2(N) → �2(N) for k ∈ N, the operator definedby τ−k(x0, x1, x2, . . . ) = (xk, xk+1, . . . ) for all x = (x0, x1, . . . ) ∈ �2(N), weget

‖a0C+a1C1+. . .+anCn‖B(�2) = sup‖x‖2≤1

‖C(a0x+a1τ−1x+. . .+anτ−nx)‖2

= sup‖x‖2≤1

‖C(a0E0+a1E1+. . .+anEn)x‖2

= ‖C(

n∑k=0

akEk

)‖B(�2)

≤ ‖C‖B(�2) · ‖Pn‖C(T) .

(Here, of course, Ek is the kth -diagonal matrix associated to E whoseentries are all equal to 1.)

Now

‖A�B‖H1(�2) = sup‖C‖B(�2)≤1

‖(A�B) ∗ C‖B(�2)

≤ ‖B‖H1(�2) · ‖fA‖A(D). �


We define now the space

H2(�2)def= {A upper triangular matrix ; lim

n→∞‖σn(A) − A‖B(�2,�∞) = 0}.

Of course A ∈ H2(�2) ∩ T if and only if fA ∈ H2(T).

Then we get

Theorem 13. (Matriceal Cauchy-Schwarz inequality) Let A ∈ T suchthat fA ∈ H2(T) and B ∈ H2(�2). Then A�B ∈ H1(�2) and

‖A�B‖H1(�2) ≤ ‖fA‖H2(T) · ‖B‖H2(�2).

Proof. Since ‖A ·B‖M(�2) ≤ ‖A‖B(�2,�∞) · ‖B‖B(�1,�2) (see Theorem 6.4,[4]), if fA = Pn , we get that ‖A�B‖M(�2) = ‖B · A‖M(�2) ≤ ‖B‖B(�2,�∞) ·‖A‖B(�1,�2) = ‖fA‖H2(T) · ‖B‖H2(�2), which proves the theorem. �

In order to state a more general Holder inequality for matrices we usethe complex interpolation for spaces of upper triangular matrices; moreprecisely we consider the spaces [A(D), H2(T)] 2

p= Hp(T), (see [5]) and

[H1(�2), H2(�2)] 2p, for 2 ≤ p ≤ ∞.

By Theorems 7 and 8 it follows that Arveson projection maps H1(�2)onto H1(�2) ∩ T = H1(T), H2(�2) onto H2(�2) ∩ T = H2(T). On theother side [H1(�2), H2(�2)] 2

p= [H2(�2), H1(�2)] p−2

pand the last space will

be denoted by Hq(�2), where p−1 + q−1 = 1. The reason for this lastnotation is that Hq(�2)∩ T = Hq(T), since, by Theorem A, it is clear that[H2(T), H1(T)]θ = [H2(�2) ∩ T , H1(�2) ∩ T ]θ = [H2(�2), H1(�2)]θ ∩ T , for0 < θ < 1.

We consider the complex interpolation norm on Hq(�2) and we call thisspace the matriceal Hardy space of order q.

Since H1(�2) is not complemented in L1(�2) we ask ourselves:

Problem. Is it Hq(�2) a Banach subspace of Lq(�2)?

Of course Hq(�2)∩T = Hq(T) is a Banach subspace of Lq(�2)∩T = Lq(T)and Hq(�2) is a linear subspace of Lq(�2), for all 1 ≤ q ≤ ∞.

Using the well-known result about complex interpolation of bilinear maps(see [6]-10.1) and theorems 12 and 13 we get a matriceal version of classicalHolder inequality for analytic functions:

Theorem 14. Let A ∈ T such that fA ∈ Hp(T), for 2 ≤ p ≤ ∞, andB ∈ Hq(�2), for p−1 + q−1 = 1. Then A�B ∈ H1(�2) and

‖A�B‖H1(�2) ≤ ‖fA‖Hp(T) · ‖B‖Hq(�2).


References

[1] J. Arazy, Some remarks on interpolation theorems and the boundness ofthe triangular projection in unitary matrix spaces, Integral EquationsOperator Theory, 1 (1978), 453–495.

[2] W. Arveson, Interpolation problems in nest algebras, J. Funct. Anal.,20 (1975), 208–253.

[3] S. Barza, L.-E. Persson and N. Popa, A matriceal analogue of Fejer’stheory, Math. Nachr., 260 (2003), 14–20.

[4] G. Bennett, Schur multipliers, Duke Math. J., 44 (1977), 603–639.[5] C. Bennet and R. Sharpley, Interpolation of operators, Academic Press,

Boston, 1988.[6] A. P. Calderon, Intermediate spaces and interpolation, the complex

method, Studia Math., 24 (1964), 113–190.[7] R. E. Edwards, Fourier Series, A Modern Introduction, Vol. I & II,

2nd edn., Springer Verlag, Berlin, 1979, 1982.[8] A. L. Shields, An analogue of a Hardy-Littlewood-Fejer inequality for

upper triangular trace class operators, Math. Z., 182 (1983), 473–484.[9] H. Triebel, Interpolation theory, function spaces, differential operators,

VEB Deutscher Verlag der Wissenschaften, Berlin, 1978.[10] K. Zhu, Operator Theory in Banach Function Spaces, Marcel Dekker,

New York, 1990.

Department of Eng. SciencesPhysics and Mathematics, University of KarlstadS-651 88 KarlstadSweden(E-mail : [email protected])

Department of MathematicsNational Technical University of Athens157 80 Zografou-Campus AthensGreece(E-mail : [email protected])

Inst. of Mathematics, Romanian AcademyBox 1-764 70700BucharestRomania(E-mail : [email protected])

(Received : September 2004 )

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