We have characterized a new type of core for double sequences, PC-core, and determined the necessary and sufficient conditions on a four-dimensional matrix A to yield PC-core{Ax}⊆α(P-core{x}) for all ℓ2∞.

1. Introduction

A double sequence x=[xjk]j,k=0∞ is said to be convergent in the Pringsheim sense or P-convergent if for every ϵ>0 there exists an N∈ℕ such that |xjk-ℓ|<ε whenever j,k>N, [1]. In this case, we write P-limx=ℓ. By c2, we mean the space of all P-convergent sequences.

A double sequence x is bounded if ‖x‖=supj,k≥0|xjk|<∞.
By ℓ∞2, we denote the space of all bounded double sequences.

Note that, in contrast to the case for single sequences, a convergent double sequence need not be bounded. So, we denote by c2∞ the space of double sequences which are bounded and convergent.

A double sequence x=[xjk] is said to converge regularly if it converges in Pringsheim's sense and, in addition, the following finite limits exist: limk→∞xjk=lj,(j=1,2,3,…),limj→∞xjk=tj,(k=1,2,3,…).
Let A=[ajkmn]j,k=0∞ be a four-dimensional infinite matrix of real numbers for all m,n=0,1,…. The sums ymn=∑j=0∞∑k=0∞ajkmnxjk
are called the A-transforms of the double sequence x=[xjk]. We say that a sequence x=[xjk] is A-summable to the limit ℓ if the A-transform of x=[xjk] exists for all m,n=0,1,… and is convergent to ℓ in the Pringsheim sense, that is, limp,q→∞∑j=0p∑k=0qajkmnxjk=ymn,limm,n→∞ymn=l.

We say that a matrix A is bounded-regular if every bounded-convergent sequence x is A-summable to the same limit and the A-transforms are also bounded. The necessary and sufficient conditions for A to be bounded-regular or RH-regular (cf., Robison [2]) are limm,n→∞ajkmn=0,(j,k=0,1,…),limm,n→∞∑j=0∞∑k=0∞ajkmn=1,limm,n→∞∑j=0∞|ajkmn|=0,(k=0,1,…),limm,n→∞∑k=0∞|ajkmn|=0,(j=0,1,…),∑j=0∞∑k=0∞|ajkmn|≤C<∞(m,n=0,1,…).
A double sequence x=[xjk] is said to be almost convergent (see [3]) to a number L if limp,q→∞sups,t≥01pq∑j=0p∑k=0qxs+j,t+k=L.
Let σ be a one-to-one mapping from ℕ into itself. The almost convergence of double sequences has been generalized to the σ-convergence in [4] as follows: limp,q→∞sups,t≥01pq∑j=0p∑k=0qxσj(s),σk(t)=l,
where σj(s)=σ(σj-1(s)). In this case, we write σ-limx=ℓ. By Vσ2, we denote the set of all σ-convergent and bounded double sequences. One can see that in contrast to the case for single sequences, a convergent double sequence need not be σ-convergent. But every bounded convergent double sequence is σ-convergent. So, c2∞⊂Vσ2⊂ℓ2∞. In the case σ(i)=i+1, σ-convergence of double sequences reduces to the almost convergence. A matrix A=[ajkmn]j,k=0∞ is said to be σ-regular if Ax∈V2σ for x=[xjk]∈c2∞ with σ-limAx=limx, and we denote this by A∈(c2∞,V2σ)reg, (see [5, 6]). Mursaleen and Mohiuddine defined and characterized σ-conservative and σ-coercive matrices for double sequences [6].

A double sequence x=[xjk] of real numbers is said to be Cesáro convergent (or C1-convergent) to a number L if and only if x∈C1, where C1={x∈l2∞:limp,q→∞Tpq(x)=L;L=C1-limx},Tpq(x)=1(p+1)(q+1)∑j=1p∑k=1qxjkmn.
We shall denote by C1 the space of Cesáro convergent (C1-convergent) double sequences.

A matrix A=(ajkmn) is said to be C1-multiplicative if Ax∈C1 for x=[xjk]∈c2∞ with C1-limAx=αlimx, and in this case we write A∈(c2∞,C1)α. Note that if α=1, then C1-multiplicative matrices are said to be C1-regular matrices.

Recall that the Knopp core (or K-core) of a real number single sequence x=(xk) is defined by the closed interval [ℓ(x),L(x)], where ℓ(x)=liminfx and L(x)=limsupx. The well-known Knopp core theorem states (cf., Maddox [7] and Knopp [8]) that in order that L(Ax)≤L(x) for every bounded real sequence x, it is necessary and sufficient that A=(ank) should be regular and limn→∞∑k=0∞|ank|=1. Patterson [9] extended this idea for double sequences by defining the Pringsheim core (or P-core) of a real bounded double sequence x=[xjk] as the closed interval [P-liminfx,P-limsupx]. Some inequalities related to the these concepts have been studied in [5, 9, 10]. Let L*(x)=limsupp,q→∞sups,t1pq∑j=0p∑k=0qxj+s,k+t,Cσ(x)=limsupp,q→∞sups,t1pq∑j=0p∑k=0qxσj(s),σk(t).

Then, MR- (Moricz-Rhoades) and σ-core of a double sequence have been introduced by the closed intervals [-L*(-x),L*(-x)] and [-Cσ(-x),Cσ(x)], and also the inequalities L(Ax)≤L*(x),L*(Ax)≤L(x),L*(Ax)≤L*(x),L(Ax)≤Cσ(x),Cσ(Ax)≤L(x)
have been studies in [3–5, 11].

In this paper, we introduce the concept of C1-multiplicative matrices and determine the necessary and sufficient conditions for a matrix A=(ajkmn) to belong to the class (c2∞,C1)α. Further we investigate the necessary and sufficient conditions for the inequality C1*(Ax)≤αL(x)
for all x∈ℓ∞2.

2. Main Results

Let us write C1*(x)=limsupp,q→∞1(p+1)(q+1)∑j=0p∑k=0qxjk.
Then, we will define the PC-core of a realvalued bounded double sequence x=[xjk] by the closed interval [-C1*(-x),C1*(x)]. Since every bounded convergent double sequence is Cesáro convergent, we have C1*(x)≤P-limsupx, and hence it follows that PC-core(x)⊆P-core(x) for a bounded double sequence x=[xjk].

Lemma 2.1 ..

A matrix A=(ajkmn) is C1-multiplicative if and only if
limp,q→∞β(j,k,p,q)=0(j,k=0,1,…),limp,q→∞∑j=0∞∑k=0∞β(j,k,p,q)=α,limp,q→∞∑j=0∞|β(j,k,p,q)|=0(k=0,1,…),limp,q→∞∑k=0∞|β(j,k,p,q)|=0(j=0,1,…),∑j=0∞∑k=0∞|ajkmn|≤C<∞,(m,n=0,1,…),
where the lim means P-lim and
β(j,k,p,q)=1(p+1)(q+1)∑j=0p∑k=0qajkmn.

Proof.

Sufficiency. Suppose that the conditions (2.2)-(2.6) hold and x=[xjk]∈c2∞ with P-limj,kxjk=L, say. So that for every ϵ>0 there exists N>0 such that |xjk|<|ℓ|+ϵ whenever j,k>N.

Then, we can write
∑j=0∞∑k=0∞β(j,k,p,q)xjk=∑j=0N∑k=0Nβ(j,k,p,q)xjk+∑j=N∞∑k=0N-1β(j,k,p,q)xjk+∑j=0N-1∑k=N∞β(j,k,p,q)xjk+∑j=N+1∞∑k=N+1∞β(j,k,p,q)xjk.
Therefore,
|∑j=0∞∑k=0∞β(j,k,p,q)xjk|≤‖x‖∑j=0N∑k=0N|β(j,k,p,q)|+‖x‖∑j=N∞∑k=0N-1|β(j,k,p,q)xjk|+‖x‖∑j=0N-1∑k=N∞|β(j,k,p,q)|+(|L|+ϵ)|∑j=N+1∞∑k=N+1∞β(j,k,p,q)|.
Letting p,q→∞ and using the conditions (2.2)–(2.6), we get
|limp,q→∞∑j=0∞∑k=0∞β(j,k,p,q)xjk|≤(|L|+ϵ)α.
Since ϵ is arbitrary, C1-limAx=αL. Hence A∈(c2∞,C1)α, that is, A is C1-multiplicative.

Necessity.

Suppose that A is C1-multiplicative. Then, by the definition, the A-transform of x exists and Ax∈C1 for each x∈c2∞. Therefore, Ax is also bounded. Then, we can write
supm,n∑j=0∞∑k=0∞|ajkmnxjk|<M<∞,
for each x∈c2∞. Now, let us define a sequence y=[yjk] by
yjk={sgnajkmn,0≤j≤r,0≤k≤r,0,otherwise,m,n=0,1,2,…. Then, the necessity of (10) follows by considering the sequence y=[yjk] in (2.11).

Also, by the assumption, we have
limp,q→∞∑j=0∞∑k=0∞β(j,k,p,q)xjk=αlimj,k→∞xjk.
Now let us define the sequence eil as follows:
eil={1,(j,k)=(i,l),0,otherwise,
and write sl=∑ieil(l∈ℕ), ri=∑leil(i∈ℕ). Then, the necessity of (2.2), (2.4), and (2.5) follows from C1-limAeil, C1-limArj and C1-limAsk, respectively.

Note that when α=1, the above theorem gives the characterization of A∈(c2∞,C1)reg. Now, we are ready to construct our main theorem.

Theorem 2.2.

For every bounded double sequence x,
C1*(Ax)≤αL(x),
or (PC-core{Ax}⊆α(P-core{x})) if and only if A is C1-multiplicative and
limsupp,q→∞∑j=0∞∑k=0∞|β(j,k,p,q)|=α.

Proof.

Necessity. Let (2.15) hold and for all x∈ℓ∞2. So, since c2∞⊂ℓ∞2, then, we get
α(-L(-x))≤-C1*(-Ax)≤C1*(Ax)≤αL(x).
That is,
αliminfx≤-C1*(-Ax)≤C1*(Ax)≤αlimsupx,
where
-C1*(-Ax)=liminfp,q→∞∑j=0∞∑k=0∞β(j,k,p,q)xjk.
By choosing x=[xjk]∈c∞2, we get from (2.17) that
-C1*(-Ax)=C1*(Ax)=C1-limAx=αlimx.
This means that A is C1-multiplicative.

By Lemma 3.1 of Patterson [9], there exists a y∈ℓ∞2 with ||y||≤1 such that
C1*(Ay)=limsupp,q→∞∑j=0∞∑k=0∞β(j,k,p,q).
If we choose y=v=[vjk], it follows
vjk={1ifj=k,0,elsewhere.
Since ∥vjk∥≤1, we have from (2.15) that
α=C1*(Av)=limsupp,q→∞∑j=0∞∑k=0∞|β(j,k,p,q)|≤αL(vjk)≤α‖v‖≤α.
This gives the necessity of (2.16).

Sufficiency.

Suppose that A is C1-regular and (2.16) holds. Let x=[xjk] be an arbitrary bounded sequence. Then, there exist M,N>0 such that xjk≤K for all j,k≥0. Now, we can write the following inequality:
|∑j=0∞∑k=0∞β(j,k,p,q)xjk|=|∑j=0∞∑k=0∞(|β(j,k,p,q)|+β(j,k,p,q)2-|β(j,k,p,q)|-β(j,k,p,q)2)xjk|≤∑j=0∞∑k=0∞|β(j,k,p,q)||xjk|+∑j=0∞∑k=0∞|(|β(j,k,p,q)|-β(j,k,p,q))xjk|≤‖x‖∑j=0M∑k=0N|β(j,k,p,q)|+‖x‖∑j=M+1∞∑k=0N|β(j,k,p,q)|+‖x‖∑j=0M∑k=N+1∞|β(j,k,p,q)|+supj,k≥M,N|xjk|∑j=M+1∞∑k=N+1∞|β(j,k,p,q)|+‖x‖∑j=0∞∑k=0∞(|β(j,k,p,q)|-β(j,k,p,q)).
Using the condition of C1-multiplicative and condition (2.16), we get
C1*(Ax)≤αL(x).
This completes the proof of the theorem.

Theorem 2.3.

For x,y∈ℓ2∞, if C1-lim|x-y|=0, then C1-core{x}=C1-core{y}.

Proof.

Since C2-lim|x-y|=0, we have C1-lim(x-y)=0 and C1-lim(-(x-y))=0. Using definition of C1-core, we take C1*(x-y)=-C1*(-(x-y))=0. Since C1* is sublinear,
0=-C1*(-(x-y))≤-C1*(-x)-C1*(y).
Therefore, C1*(y)≤-C1*(-x). Since -C1*(-x)≤C1*(x), this implies that C1*(y)≤C1*(x).By an argument similar as above, we can show that C1*(x)≤C1*(y). This completes the proof.

Acknowledgment

The authors would like to state their deep thanks to the referees for their valuable suggestions improving the paper.

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