We examine some properties of the invariant mean, define the concepts of strong
σ-convergence and absolute
σ-convergence for double sequences, and determine
the associated sublinear functionals. We also define the absolute invariant mean
through which the space of absolutely σ-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|<∞. The space of all bounded double sequences is denoted
by ℳu.

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 ɛ>0 there exists an integer N such that |xjk-L|<ɛ whenever j,k>N. In this case L is called the p-limit of x. If in addition x∈ℳu, then x is said to 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:=limkxjk(j∈ℕ) and xk:=limjxjk(k∈ℕ) exist. Note that in this case the limits
limjlimkxjk and limklimjxjk exist and are equal to the
p-limit of x.

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

Let Ω denote the vector space of all double sequences
with the vector space operations defined coordinatewise. Vector subspaces of
Ω are called double sequence
spaces.

All considered double sequence spaces are supposed to containspan{ejk∣j,k∈N}, whereeiljk={1,if(j,k)=(i,l),0,otherwise.

We denote the pointwise sums ∑j,kejk, ∑jejk(k∈ℕ), and ∑kejk, (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
setFA(ν):={x∈Ω∣[Ax]mn:=ν-∑j,kamnjkxjkexistsandAx:=([Ax]mn)m,n∈F}. Then we say that A maps the space E into the space F if E⊂FA(ν) and denote by (E,F) the set of all 4-dimensional matrices
A which map E into F.

We say that a 4-dimensional matrix A is 𝒞ν-conservative if
𝒞ν⊂𝒞νA(ν), and 𝒞ν-regular if in
additionν-limAx:=ν-limm,n[Ax]mn=ν-limm,nxmn(x∈Cν), whereCνA(ν):={x∈Ω∣[Ax]mn:=ν-∑j,kamnjkxjkexistsandAx:=([Ax]mn)m,n∈Cν}.

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 continuous linear 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-means φ. We denote by Vσ the set of all σ-convergent sequences x=(xk). Clearly c⊂Vσ. Note that a σ-mean extends the limit functional on
c in the sense that φ(x)=limx for all x∈c if and only 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 only if (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 all j,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, whereVσ={x∈Mu:limp,q→∞τpqst(x)=Luniformlyins,t;L=σ-limx},τpqst:=τpqst(x)=1(p+1)(q+1)∑j=0p∑k=0qxσj(s),σk(t).τ0qst:=τ0qst(x)=1(q+1)∑k=0qxs,σk(t),τp0st:=τp0st(x)=1(p+1)∑j=0pxσj(s),t,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 to a number L if and only iflimp,q→∞1(p+1)(q+1)∑j=0p∑k=0qxj+s,k+t=Luniformlyins,t. 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 exclude
the possibility that every boundedly convergent double sequence might have a
uniquely determined 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 double
sequence (and hence, in particular, any boundedly convergent double sequence)
has σ-mean x00.

In this paper we examine some properties of the invariant mean and define the
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
invariant mean through which the space of absolutely σ-convergent double sequences is
characterized.

2. Strong and Absolute <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M134"><mml:mrow><mml:mi>σ</mml:mi></mml:mrow></mml:math></inline-formula>-Convergence

In this section we define the concepts of strong σ-convergence and absolute σ-convergence for 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 dominates and generates the
σ-means, where V:ℳu→ℝ is defined by V(x)=infp=(pjk)∈V0σlimsupj,k(xjk+pjk).

Now we investigate the sublinear functional which generates the space
[𝒱σ] of strongly σ-convergent double sequences defined in [22] as [Vσ]={x=(xjk)∈Mu:limp,q→∞1(p+1)(q+1)∑j=0p∑k=0q|xσj(s),σk(t)-L|=0,uniformlyins,t}.

Definition 2.2.

We define Ψ:ℳu→ℝ by Ψ(x)=limsupp,qsups,t1(p+1)(q+1)∑j=0p∑k=0q|xσj(s),σk(t)|.

Let {ℳu,Ψ} denote the set of all linear functionals
Φ on ℳu such that Φ(x)≤Ψ(x) for all 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}-limx=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 all s,t, 1(p+1)(q+1)∑j=0p∑k=0q|xσj(s),σk(t)-L|<ϵ, 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)=0for all Φ∈{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}.

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), it follows that x∈[𝒱σ]. This completes the proof of the theorem.

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

Putϕpqst(x)=τpqst(x)-τp-1,q,s,t(x)-τp,q-1,s,t(x)+τp-1,q-1,s,t(x). Thus simplifying further, we getϕpqst(x)=1p(p+1)∑m=1pm[1q(q+1)∑n=1qn(xσm(s),σn(t)-xσm(s),σn-1(t))]=1p(p+1)q(q+1)×∑m=1p∑n=1qmn[xσm(s),σn(t)-xσm-1(s),σn(t)-xσm(s),σn-1(t)+xσm-1(s),σn-1(t)]. Now we writeϕpqst(x)={1p(p+1)q(q+1)×∑m=1p∑n=1qmn[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)∑n=1qn[xs,σn(t)-xs,σn-1(t)],p=0,q≥1,1p(p+1)∑n=1pm[xσm(s),t-xσm(s),t],p≥1,q=0, and ϕ00st(x)=xst.

In [9], the following was defined.

Definition 2.4.

A double sequence x=(xjk)∈ℳu is said to be
absolutelyσ-almost convergent if and
only if ∑p=0∞∑q=0∞|ϕpqst(x)|convergesuniformlyins,t.

By 𝒲σ, 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 and
only if

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

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 is easy to
prove that both 𝒲σ and ℬ𝒱σ are Banach spaces with the norm ‖x‖=sups,t∑p=0∞∑q=0∞|ϕpqst(x)|.

Note that ℬ𝒱σ⊂𝒲σ⊂𝒱σ.

Remark 2.6.

It is easy to see that the assertion (i) implies that (τpqst(x)) (as a double sequence in
p,q) converges uniformly in
s,t, but it may converge to a different limit for
different values of s,t. This point did not arise in Banach limit case
in which σ(n)=n+1. In this case if we assume only that
limp,q→∞τpqst(x)=ℓ for some value of s,t; then we must have limp,q→∞τpqst(x)=ℓ for any other s,t (but not necessarily uniformly in
s,t). So if, as a special case, we assume uniform
convergence, the value to τpqst(x) converges must be same for all
s,t. This need not be in the general case. For
example, consider σ(n)=n+2. Define the sequence x=(xjk) by xjk={1,ifjisodd,∀k,0,ifjiseven,∀k. Then for all p,q≥0τpqst(x)={1,ifsisodd,∀t,0,ifsiseven,∀t,0,otherwise, 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 is proper.

3. Absolute Invariant MeanRemark 3.1.

It may be remarked that we have a class of linear continuous functionals
φ2 on ℳu (which we call the set of invariant means) such
that φ2 is uniquely determined if and only if
x∈𝒱σ, that is, the largest set which determines
φ2 uniquely is 𝒱σ. Now we are going to 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 on ℳu such that φ2 is uniquely determined if and only if
x∈ℬ𝒱σ.

Proof.

We first note that ℬ𝒱σ is not closed in ℳu (which follows from the case
σ(n)=n+1 for single sequences which is proved in [23]). Given the value of
φ2(x) for x∈ℬ𝒱σ, its value for x∈cl(ℬ𝒱σ) is determined by continuity. So if
φ2(x) is unique for x∈ℬ𝒱σ, it must be unique in 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)=limsupp,qsups,tτpqst(x) both dominates and generates the functional
φ2 which is a σ-mean if and only if -λ(-x)≤φ2(x)≤λ(x). It follows from (3.2) that φ2 is unique σ-mean if and only if Vσ={x∈Mu:λ(x)=-λ(-x)}.

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 on ℳu is an absolute invariant mean
(𝒜ℐℳ) if and only if -Q(-x)≤ψ2(x)≤Q(x) and is unique 𝒜ℐℳ if and only if cl(BVσ)={x∈Mu:Q(x)=-Q(-x)}, where Q(x)=limsupp,qsups,t∑i=p∞∑j=q∞|ϕijst(x)|<∞.

We have the following result.

Theorem 3.5.

One has BVσ={x∈Mu:Q(x)=0}.

Proof.

Since Q is a sublinear functional on
ℳu, it follows from Hahn-Banach Theorem that there
exists a continuous linear functional μ on ℳu such that μ(x)≤Q(x)∀x∈Mu, and this limit is unique if and only if
Q(x)=-Q(-x)=-Q(x), that is, if and only if
Q(x)=0 for all x∈ℳu. That is, if and only if limp,q∑i=p∞∑j=q∞|ϕijst(x)|=0uniformlyins,t, that is, if and only if x∈ℬ𝒱σ.

Acknowledgments

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

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