invariant and absolute invariant means of double...

Hindawi Publishing CorporationJournal of Function Spaces and ApplicationsVolume 2012, Article ID 465364, 9 pagesdoi:10.1155/2012/465364

Research ArticleInvariant and Absolute InvariantMeans of Double Sequences

Abdullah Alotaibi,1 M. Mursaleen,2 and M. A. Alghamdi1

1 Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia2 Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

Correspondence should be addressed to M. Mursaleen, [email protected]

Received 20 February 2012; Revised 17 April 2012; Accepted 28 April 2012

Academic Editor: Sivaram K. Narayan

Copyright q 2012 Abdullah Alotaibi et al. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We examine some properties of the invariant mean, define the concepts of strong σ-convergenceand absolute σ-convergence for double sequences, and determine the associated sublinearfunctionals. We also define the absolute invariant mean through which the space of absolutelyσ-convergent double sequences is characterized.

1. Introduction and Preliminaries

For the following notions, we refer to [1, 2].A double sequence x = (xjk) of real or complex numbers is said to be bounded if

‖x‖∞ = supj,k

∣∣xjk

∣∣ <∞. (1.1)

The space of all bounded double sequences is denoted byMu.A double sequence x = (xjk) is said to converge to the limit L in Pringsheim’s sense

(shortly, p-convergent to L) if for every ε > 0 there exists an integer N such that |xjk − L| < εwhenever j, k > N. In this case L is called the p-limit of x. If in addition x ∈ Mu, then x is saidto be boundedly convergent to L in Pringsheim’s sense (shortly, bp-convergent to L).

A double sequence x = (xjk) is said to converge regularly to L (shortly, r-convergent to L)if x is p-convergent and the limits xj := limk xjk (j ∈ N) and xk := limj xjk (k ∈ N) exist. Notethat in this case the limits limj limk xjk and limk limj xjk exist and are equal to the p-limit ofx.

2 Journal of Function Spaces and Applications

In general, for any notion of convergence ν, the space of all ν-convergent doublesequences will be denoted by Cν and the limit of a ν-convergent double sequence x byν-limj,k xjk, where ν ∈ {p, bp, r}.

LetΩ denote the vector space of all double sequences with the vector space operationsdefined coordinatewise. Vector subspaces of Ω are called double sequence spaces.

All considered double sequence spaces are supposed to contain

span{

ejk | j, k ∈ N

}

, (1.2)

where

ejkil =

⎧

⎨

⎩

1, if(

j, k)

= (i, �),

0, otherwise.(1.3)

We denote the pointwise sums∑

j,k ejk,

∑

j ejk (k ∈ N), and

∑

k ejk, (j ∈ N) by e, ek and

ej, respectively.Let E be the space of double sequences converging with respect to a convergence

notion ν, F a double sequence space, and A = (amnjk) a 4-dimensional matrix of scalars.Define the set

F(ν)A :=

⎧

⎨

⎩x ∈ Ω | [Ax]mn := ν-

∑

j,k

amnjkxjk exists and Ax := ([Ax]mn)m,n ∈ F⎫

⎬

⎭. (1.4)

Then we say that Amaps the space E into the space F if E ⊂ F(ν)A and denote by (E, F) the set

of all 4-dimensional matrices Awhich map E into F.We say that a 4-dimensional matrix A is Cν-conservative if Cν ⊂ C(ν)

νA, and Cν-regular ifin addition

ν- limAx := ν-limm,n

[Ax]mn = ν-limm,n

xmn (x ∈ Cν), (1.5)

where

C(ν)νA :=

⎧

⎨

⎩x ∈ Ω | [Ax]mn := ν-

∑

j,k

amnjkxjk exists and Ax := ([Ax]mn)m,n ∈ Cν

⎫

⎬

⎭. (1.6)

Matrix transformations for double sequences are considered by various authors,namely, [3–5].

Let σ be a one-to-one mapping from the set N0 = {0, 1, 2, . . . .} into itself. A continuouslinear functional ϕ on l∞ is said to be an invariant mean or a σ-mean (see [6, 7]) if and only if (i)ϕ(x) ≥ 0 when the sequence x = (xk) has xk ≥ 0 for all k, (ii) ϕ(e) = 1, where e = (1, 1, 1, . . .),and (iii) ϕ(x) = ϕ(xσ(k)) for all x ∈ l∞.

We say that a sequence x = (xk) is σ-convergent to the limit L if ϕ(x) = L for all σ-meansϕ. We denote by Vσ the set of all σ-convergent sequences x = (xk). Clearly c ⊂ Vσ . Note that

Journal of Function Spaces and Applications 3

a σ-mean extends the limit functional on c in the sense that ϕ(x) = lim x for all x ∈ c if andonly if σ has no finite orbits, that is to say, if and only if σk(n)/=n, for all n ≥ 0, k ≥ 1 (see[8]).

Recently, the concept of invariant mean for double sequences was defined in [9].Let σ be a one-to-one mapping from the set N of natural numbers into itself. A

continuous linear functional ϕ2 on Mu is said to be an invariant mean or a σ-mean if and onlyif (i) ϕ2(x) ≥ 0 if x ≥ 0 (i.e., xjk ≥ 0 for all j, k); (ii) ϕ2(E) = 1, where E = (ejk), ejk = 1 for allj, k, and (iii) ϕ2(x) = ϕ2((xσ(j),σ(k))) = ϕ2((xσ(j),k)) = ϕ2((xj,σ(k))).

If σ(n) = n + 1 then σ-mean is reduced to the Banach limit for double sequences [10].The idea of σ-convergence for double sequences has recently been introduced in [11]

and further studied in [9, 12–16].A double sequence x = (xjk) of real numbers is said to be σ-convergent to a number L

if and only if x ∈ Vσ , where

Vσ ={

x ∈ Mu : limp, q→∞

τpqst(x) = L uniformly in s, t; L = σ − lim x

}

,

τpqst := τpqst(x) =1

(

p + 1)(

q + 1)

p∑

j=0

q∑

k=0

xσj (s),σk(t).

τ0qst := τ0qst(x) =1

(

q + 1)

q∑

k=0

xs,σk(t),

τp0st := τp0st(x) =1

(

p + 1)

p∑

j=0xσj (s),t,

(1.7)

τ0,0,s,t = xst and τ−1,q,s,t = τp,−1,s,t = τ−1,−1,s,t = 0.Note that Cbp ⊂ Vσ ⊂ Mu.Throughout this paper limit of a double sequence means bp-limit.For σ(n) = n + 1, the set Vσ is reduced to the set f2 of almost convergent double

sequences [17]. A double sequence x = (xjk) of real numbers is said to be almost convergent toa number L if and only if

limp,q→∞

1(

p + 1)(

q + 1)

p∑

j=0

q∑

k=0

xj+s,k+t = L uniformly in s, t. (1.8)

The concept of almost convergence for single sequences was introduced by Lorentz [18].

Remark 1.1. In view of the following example, it may be remarked that this does not excludethe possibility that every boundedly convergent double sequence might have a uniquelydetermined σ-mean not necessarily equal to its bp-limit.

For example, let σ(n) = 0 for all n. Then it is easily seen that any bounded doublesequence (and hence, in particular, any boundedly convergent double sequence) has σ-meanx00.

In this paper we examine some properties of the invariant mean and definethe concepts of absolute σ-convergence and strong σ-convergence for double sequences


analogous to the case of single sequences [8, 19]. We further define the absolute invariantmean through which the space of absolutely σ-convergent double sequences is characterized.

2. Strong and Absolute σ-Convergence

In this section we define the concepts of strong σ-convergence and absolute σ-convergencefor double sequences. These concepts for single sequences were studied in [8, 19–21].

Remark 2.1. In [9], it was shown that the sublinear functional V defined on Mu dominatesand generates the σ-means, where V : Mu → R is defined by

V (x) = infp=(pjk)∈V0σ

lim supj,k

(

xjk + pjk)

. (2.1)

Now we investigate the sublinear functional which generates the space [Vσ] ofstrongly σ-convergent double sequences defined in [22] as

[Vσ] =

⎧

⎨

⎩x =

(

xjk) ∈ Mu : lim

p,q→∞1

(

p + 1)(

q + 1)

p∑

j=0

q∑

k=0

∣∣xσj (s),σk(t) − L

∣∣ = 0, uniformly in s, t

⎫

⎬

⎭.

(2.2)

Definition 2.2. We define Ψ : Mu → R by

Ψ(x) = lim supp,q

sups,t

1(

p + 1)(

q + 1)

p∑

j=0

q∑

k=0

∣∣xσj (s),σk(t)

∣∣. (2.3)

Let {Mu,Ψ} denote the set of all linear functionals Φ onMu such that Φ(x) ≤ Ψ(x) forall x = (xjk) ∈ Mu. By Hahn-Banach Theorem, the set {Mu,Ψ} is nonempty.

If there exists L ∈ R such that

Φ(x − Le) = 0 ∀Φ ∈ {Mu,Ψ}, (∗)

then we say that x is {Mu,Ψ}-convergent to L and in this case we write {Mu,Ψ}-lim x = L.

We are now ready to prove the following result.

Theorem 2.3. [Vσ] is the set of all {Mu,Ψ}-convergent sequences.

Proof. Let x ∈ [Vσ]. Then for each ε > 0, there exist p0, q0 such that for p > p0, q > q0 and alls, t,

1(

p + 1)(

q + 1)

p∑

j=0

q∑

k=0

∣∣xσj (s),σk(t) − L

∣∣ < ε, (2.4)


and this implies that Ψ(x − Le) ≤ ε. In a similar manner, we can prove that Ψ(Le − x) ≤ ε.Hence |Φ(x − Le)| ≤ Ψ(x − Le) ≤ ε for all Φ ∈ {Mu,Ψ}. Therefore Φ(x − Le) = 0 for allΦ ∈ {Mu,Ψ} and this implies that by (2.6) x ∈ [Vσ] implies that x is {Mu,Ψ}-convergent.

Conversely, suppose that x is {Mu,Ψ}-convergent, that is,

Φ(x − Le) = 0 ∀Φ ∈ {Mu,Ψ}. (2.5)

Since Ψ is sublinear functional on Mu, by Hahn-Banach Theorem, there exists Φ0 ∈{Mu,Ψ} such that Φ0(x − Le) = Ψ(x − Le). Hence Ψ(x − Le) = 0; since Ψ(x) = Ψ(−x), itfollows that x ∈ [Vσ]. This completes the proof of the theorem.

Now we define the concept of absolute σ-convergence for double sequences.Put

φpqst(x) = τpqst(x) − τp−1,q,s,t(x) − τp,q−1,s,t(x) + τp−1,q−1,s,t(x). (2.6)

Thus simplifying further, we get

φpqst(x) =1

p(

p + 1)

p∑

m=1

m

[

1q(

q + 1)

q∑

n=1

n(

xσm(s),σn(t) − xσm(s),σn−1(t))

]

=1

p(

p + 1)

q(

q + 1)

×p∑

m=1

q∑

n=1

mn[

xσm(s),σn(t)− xσm−1(s),σn(t)−xσm(s),σn−1(t)+ xσm−1(s),σn−1(t)]

.

(2.7)

Now we write

φpqst(x) =

⎧

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1p(

p + 1)

q(

q + 1)

×p∑

m=1

q∑

n=1

mn[

xσm(s),σn(t)− xσm−1(s),σn(t)−xσm(s),σn−1(t)+ xσm−1(s),σn−1(t)]

, p, q ≥ 1,

1q(

q + 1)

q∑

n=1

n[

xs,σn(t) − xs,σn−1(t)]

, p = 0, q ≥ 1,

1p(

p + 1)

p∑

n=1

m[

xσm(s),t − xσm(s),t]

, p ≥ 1, q = 0,

(2.8)

and φ00st(x) = xst.In [9], the following was defined.


Definition 2.4. A double sequence x = (xjk) ∈ Mu is said to be absolutely σ-almost convergent ifand only if

∞∑

p=0

∞∑

q=0

∣∣φpqst(x)

∣∣ converges uniformly in s, t. (2.9)

ByWσ , we denote the space of all absolutely σ-almost convergent double sequences.

Now we define the following.

Definition 2.5. A double sequence x = (xjk) ∈ Mu is said to be absolutely σ-convergent if andonly if

(i)∑∞

p=0∑∞

q=0 |φpqst(x)| converges uniformly in s, t;

(ii) limp,q→∞τpqst(x), which must exist, should take the same value for all s, t.

By BVσ , we denote the space of all absolutely σ-convergent double sequences. It iseasy to prove that bothWσ and BVσ are Banach spaces with the norm

‖x‖ = sups,t

∞∑

p=0

∞∑

q=0

∣∣φpqst(x)

∣∣. (2.10)

Note that BVσ ⊂ Wσ ⊂ Vσ .

Remark 2.6. It is easy to see that the assertion (i) implies that (τpqst(x)) (as a double sequencein p, q) converges uniformly in s, t, but it may converge to a different limit for differentvalues of s, t. This point did not arise in Banach limit case in which σ(n) = n + 1. In thiscase if we assume only that limp,q→∞ τpqst(x) = � for some value of s, t; then we must havelimp,q→∞ τpqst(x) = � for any other s, t (but not necessarily uniformly in s, t). So if, as a specialcase, we assume uniform convergence, the value to τpqst(x) converges must be same for alls, t. This need not be in the general case. For example, consider σ(n) = n + 2. Define thesequence x = (xjk) by

xjk =

⎧

⎨

⎩

1, if j is odd, ∀k,0, if j is even, ∀k.

(2.11)

Then for all p, q ≥ 0

τpqst(x) =

⎧

⎪⎪⎪⎨

⎪⎪⎪⎩

1, if s is odd, ∀t,0, if s is even, ∀t,0, otherwise,

(2.12)

so that φpqst(x) = 0 for all p, q ≥ 1 (in particular, φ1111(x) = xσ(1),σ(1)−x11 = x3,3−x11 = 1−1 = 0,since σ(1) = 1 + 2 = 3). Thus (i) certainly holds, but the value of limp,q→∞ τpqst(x) is 1 when


s is odd and 0 when s is even (for all t). Moreover, it shows that the inclusion BVσ ⊂ Wσ isproper.

3. Absolute Invariant Mean

Remark 3.1. It may be remarked that we have a class of linear continuous functionals ϕ2 onMu (which we call the set of invariant means) such that ϕ2 is uniquely determined if andonly if x ∈ Vσ , that is, the largest set which determines ϕ2 uniquely is Vσ . Now we are goingto deal with the similar situation which prevails for BVσ .

As an immediate consequence, we have the following.

Theorem 3.2. There does not exist a class of continuous linear functionals ϕ2 onMu such thatϕ2 is uniquely determined if and only if x ∈ BVσ .

Proof. We first note that BVσ is not closed inMu (which follows from the case σ(n) = n+1 forsingle sequences which is proved in [23]). Given the value of ϕ2(x) for x ∈ BVσ , its value forx ∈ cl(BVσ) is determined by continuity. So if ϕ2(x) is unique for x ∈ BVσ , it must be uniquein the set cl(BVσ), which is larger than BVσ .

Remark 3.3. As in Remark 2.1, it is easy to see that the sublinear functional

λ(x) = lim supp,q

sups,t

τpqst(x) (3.1)

both dominates and generates the functional ϕ2 which is a σ-mean if and only if

−λ(−x) ≤ ϕ2(x) ≤ λ(x). (3.2)

It follows from (3.2) that ϕ2 is unique σ-mean if and only if

Vσ = {x ∈ Mu : λ(x) = −λ(−x)}. (3.3)

In the same vein, we seek a characterization of a class of linear functionals ψ2 on Mu

to define absolute invariant mean in terms of a suitable sublinear functional Q on Mu.

Definition 3.4. A linear functional ψ2 onMu is an absolute invariant mean (AIM) if and onlyif −Q(−x) ≤ ψ2(x) ≤ Q(x) and is unique AIM if and only if

cl(BVσ) = {x ∈ Mu : Q(x) = −Q(−x)}, (3.4)

where

Q(x) = lim supp,q

sups,t

∞∑

i=p

∞∑

j=q

∣∣φijst(x)

∣∣ <∞. (3.5)

We have the following result.


Theorem 3.5. One has

BVσ = {x ∈ Mu : Q(x) = 0}. (3.6)

Proof. Since Q is a sublinear functional on Mu, it follows from Hahn-Banach Theorem thatthere exists a continuous linear functional μ on Mu such that

μ(x) ≤ Q(x) ∀x ∈ Mu, (3.7)

and this limit is unique if and only if Q(x) = −Q(−x) = −Q(x), that is, if and only if Q(x) = 0for all x ∈ Mu. That is, if and only if

limp,q

∞∑

i=p

∞∑

j=q

∣∣φijst(x)

∣∣ = 0 uniformly in s, t, (3.8)

that is, if and only if x ∈ BVσ .

Acknowledgments

The authors would like to thank the Deanship of Scientific Research at King AbdulazizUniversity for its financial support under Grant no. 99-130-1432. The authors are also thankfulto the referee for his/her constructive comments which helped to improve the present paper.

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