Abstract

We have characterized a new type of core for double sequences, š‘ƒš¶-core, and determined the necessary and sufficient conditions on a four-dimensional matrix š“ to yield š‘ƒš¶-core{š“š‘„}āŠ†š›¼(š‘ƒ-core{š‘„}) for all ā„“āˆž2.

1. Introduction

A double sequence š‘„=[š‘„š‘—š‘˜]āˆžš‘—,š‘˜=0 is said to be convergent in the Pringsheim sense or š‘ƒ-convergent if for every šœ–>0 there exists an š‘āˆˆā„• such that |š‘„š‘—š‘˜āˆ’ā„“|<šœ€ whenever š‘—,š‘˜>š‘, [1]. In this case, we write š‘ƒāˆ’limš‘„=ā„“. By š‘2, we mean the space of all š‘ƒ-convergent sequences.

A double sequence š‘„ is bounded if ā€–š‘„ā€–=supš‘—,š‘˜ā‰„0||š‘„š‘—š‘˜||<āˆž.(1.1) By ā„“2āˆž, 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 š‘āˆž2 the space of double sequences which are bounded and convergent.

A double sequence š‘„=[š‘„š‘—š‘˜] is said to converge regularly if it converges in Pringsheim's sense and, in addition, the following finite limits exist: limš‘˜ā†’āˆžš‘„š‘—š‘˜=ā„“š‘—,(š‘—=1,2,3,ā€¦),limš‘—ā†’āˆžš‘„š‘—š‘˜=š‘”š‘—,(š‘˜=1,2,3,ā€¦).(1.2) Let š“=[š‘Žš‘šš‘›š‘—š‘˜]āˆžš‘—,š‘˜=0 be a four-dimensional infinite matrix of real numbers for all š‘š,š‘›=0,1,ā€¦. The sums š‘¦š‘šš‘›=āˆžī“āˆžš‘—=0ī“š‘˜=0š‘Žš‘šš‘›š‘—š‘˜š‘„š‘—š‘˜(1.3) are called the š“-transforms of the double sequence š‘„=[š‘„š‘—š‘˜]. We say that a sequence š‘„=[š‘„š‘—š‘˜] is š“-summable to the limit ā„“ if the š“-transform of š‘„=[š‘„š‘—š‘˜] exists for all š‘š,š‘›=0,1,ā€¦ and is convergent to ā„“ in the Pringsheim sense, that is, limš‘š‘,š‘žā†’āˆžī“š‘žš‘—=0ī“š‘˜=0š‘Žš‘šš‘›š‘—š‘˜š‘„š‘—š‘˜=š‘¦š‘šš‘›,limš‘š,š‘›ā†’āˆžš‘¦š‘šš‘›=ā„“.(1.4)

We say that a matrix š“ is bounded-regular if every bounded-convergent sequence š‘„ is š“-summable to the same limit and the š“-transforms are also bounded. The necessary and sufficient conditions for š“ to be bounded-regular or RH-regular (cf., Robison [2]) are limš‘š,š‘›ā†’āˆžš‘Žš‘šš‘›š‘—š‘˜=0,(š‘—,š‘˜=0,1,ā€¦),limāˆžš‘š,š‘›ā†’āˆžī“āˆžš‘—=0ī“š‘˜=0š‘Žš‘šš‘›š‘—š‘˜=1,limāˆžš‘š,š‘›ā†’āˆžī“š‘—=0||š‘Žš‘šš‘›š‘—š‘˜||=0,(š‘˜=0,1,ā€¦),limāˆžš‘š,š‘›ā†’āˆžī“š‘˜=0||š‘Žš‘šš‘›š‘—š‘˜||=0,(š‘—=0,1,ā€¦),āˆžāˆ‘āˆžš‘—=0āˆ‘š‘˜=0||š‘Žš‘šš‘›š‘—š‘˜||.ā‰¤š¶<āˆž(š‘š,š‘›=0,1,ā€¦)(1.5) A double sequence š‘„=[š‘„š‘—š‘˜] is said to be almost convergent (see [3]) to a number šæ if limš‘,š‘žā†’āˆžsupš‘ ,š‘”ā‰„01š‘š‘žš‘ī“š‘žš‘—=0ī“š‘˜=0š‘„š‘ +š‘—,š‘”+š‘˜=šæ.(1.6) Let šœŽ be a one-to-one mapping from ā„• into itself. The almost convergence of double sequences has been generalized to the šœŽ-convergence in [4] as follows: limš‘,š‘žā†’āˆžsupš‘ ,š‘”ā‰„01š‘š‘žš‘ī“š‘žš‘—=0ī“š‘˜=0š‘„šœŽš‘—(š‘ ),šœŽš‘˜(š‘”)=ā„“,(1.7) where šœŽš‘—(š‘ )=šœŽ(šœŽš‘—āˆ’1(š‘ )). In this case, we write šœŽāˆ’limš‘„=ā„“. By š‘‰2šœŽ, we denote the set of all šœŽ-convergent and bounded double sequences. One can see that in contrast to the case for single sequences, a convergent double sequence need not be šœŽ-convergent. But every bounded convergent double sequence is šœŽ-convergent. So, š‘āˆž2āŠ‚š‘‰2šœŽāŠ‚ā„“āˆž2. In the case šœŽ(š‘–)=š‘–+1, šœŽ-convergence of double sequences reduces to the almost convergence. A matrix š“=[š‘Žš‘šš‘›š‘—š‘˜]āˆžš‘—,š‘˜=0 is said to be šœŽ-regular if š“š‘„āˆˆš‘‰šœŽ2 for š‘„=[š‘„š‘—š‘˜]āˆˆš‘āˆž2 with šœŽāˆ’limš“š‘„=limš‘„, and we denote this by š“āˆˆ(š‘āˆž2,š‘‰šœŽ2)reg, (see [5, 6]). Mursaleen and Mohiuddine defined and characterized šœŽ-conservative and šœŽ-coercive matrices for double sequences [6].

A double sequence š‘„=[š‘„š‘—š‘˜] of real numbers is said to be CesĆ”ro convergent (or š¶1-convergent) to a number L if and only if š‘„āˆˆš¶1, where š¶1=ī‚»š‘„āˆˆā„“āˆž2āˆ¶limš‘,š‘žā†’āˆžš‘‡š‘š‘ž(š‘„)=šæ;šæ=š¶1ī‚¼,š‘‡āˆ’limš‘„š‘š‘ž(1š‘„)=(š‘+1)(š‘ž+1)š‘ī“š‘žš‘—=1ī“š‘˜=1š‘„š‘šš‘›š‘—š‘˜.(1.8) We shall denote by š¶1 the space of CesĆ”ro convergent (š¶1-convergent) double sequences.

A matrix š“=(š‘Žš‘šš‘›š‘—š‘˜) is said to be š¶1-multiplicative if š“š‘„āˆˆš¶1 for š‘„=[š‘„š‘—š‘˜]āˆˆš‘āˆž2 with š¶1āˆ’limš“š‘„=š›¼limš‘„, and in this case we write š“āˆˆ(š‘āˆž2,š¶1)š›¼. Note that if š›¼=1, then š¶1-multiplicative matrices are said to be š¶1-regular matrices.

Recall that the Knopp core (or K-core) of a real number single sequence š‘„=(š‘„š‘˜) is defined by the closed interval [ā„“(š‘„),šæ(š‘„)], where ā„“(š‘„)=liminfš‘„ and šæ(š‘„)=limsupš‘„. The well-known Knopp core theorem states (cf., Maddox [7] and Knopp [8]) that in order that šæ(š“š‘„)ā‰¤šæ(š‘„) for every bounded real sequence š‘„, it is necessary and sufficient that š“=(š‘Žš‘›š‘˜) should be regular and limš‘›ā†’āˆžāˆ‘āˆžš‘˜=0|š‘Žš‘›š‘˜|=1. Patterson [9] extended this idea for double sequences by defining the Pringsheim core (or P-core) of a real bounded double sequence š‘„=[š‘„š‘—š‘˜] as the closed interval [š‘ƒāˆ’liminfš‘„,š‘ƒāˆ’limsupš‘„]. Some inequalities related to the these concepts have been studied in [5, 9, 10]. Let šæāˆ—(š‘„)=limsupš‘,š‘žā†’āˆžsupš‘ ,š‘”1š‘š‘žš‘ī“š‘žš‘—=0ī“š‘˜=0š‘„š‘—+š‘ ,š‘˜+š‘”,š¶šœŽ(š‘„)=limsupš‘,š‘žā†’āˆžsupš‘ ,š‘”1š‘š‘žš‘ī“š‘žš‘—=0ī“š‘˜=0š‘„šœŽš‘—(š‘ ),šœŽš‘˜(š‘”).(1.9)

Then, MR- (Moricz-Rhoades) and šœŽ-core of a double sequence have been introduced by the closed intervals [āˆ’šæāˆ—(āˆ’š‘„),šæāˆ—(āˆ’š‘„)] and [āˆ’š¶šœŽ(āˆ’š‘„),š¶šœŽ(š‘„)], and also the inequalities šæ(š“š‘„)ā‰¤šæāˆ—(š‘„),šæāˆ—(š“š‘„)ā‰¤šæ(š‘„),šæāˆ—(š“š‘„)ā‰¤šæāˆ—(š‘„),šæ(š“š‘„)ā‰¤š¶šœŽ(š‘„),š¶šœŽ(š“š‘„)ā‰¤šæ(š‘„)(1.10) have been studies in [3ā€“5, 11].

In this paper, we introduce the concept of š¶1-multiplicative matrices and determine the necessary and sufficient conditions for a matrix š“=(š‘Žš‘šš‘›š‘—š‘˜) to belong to the class (š‘āˆž2,š¶1)š›¼. Further we investigate the necessary and sufficient conditions for the inequality š¶āˆ—1(š“š‘„)ā‰¤š›¼šæ(š‘„)(1.11) for all š‘„āˆˆā„“2āˆž.

2. Main Results

Let us write š¶āˆ—1(š‘„)=limsupš‘,š‘žā†’āˆž1(š‘+1)(š‘ž+1)š‘ī“š‘žš‘—=0ī“š‘˜=0š‘„š‘—š‘˜.(2.1) Then, we will define the š‘ƒš¶-core of a realvalued bounded double sequence š‘„=[š‘„š‘—š‘˜] by the closed interval [āˆ’š¶āˆ—1(āˆ’š‘„),š¶āˆ—1(š‘„)]. Since every bounded convergent double sequence is CesĆ”ro convergent, we have š¶āˆ—1(š‘„)ā‰¤š‘ƒāˆ’limsupš‘„, and hence it follows that š‘ƒš¶-core(š‘„)āŠ†š‘ƒ-core(š‘„) for a bounded double sequence š‘„=[š‘„š‘—š‘˜].

Lemma 2.1 .. A matrix š“=(š‘Žš‘šš‘›š‘—š‘˜) is š¶1-multiplicative if and only if limš‘,š‘žā†’āˆžš›½(š‘—,š‘˜,š‘,š‘ž)=0(š‘—,š‘˜=0,1,ā€¦),(2.2)limāˆžš‘,š‘žā†’āˆžī“āˆžš‘—=0ī“š‘˜=0š›½(š‘—,š‘˜,š‘,š‘ž)=š›¼,(2.3)limāˆžš‘,š‘žā†’āˆžī“š‘—=0||||š›½(š‘—,š‘˜,š‘,š‘ž)=0(š‘˜=0,1,ā€¦),(2.4)limāˆžš‘,š‘žā†’āˆžī“š‘˜=0||||š›½(š‘—,š‘˜,š‘,š‘ž)=0(š‘—=0,1,ā€¦),(2.5)āˆžī“āˆžš‘—=0ī“š‘˜=0||š‘Žš‘šš‘›š‘—š‘˜||ā‰¤š¶<āˆž,(š‘š,š‘›=0,1,ā€¦),(2.6) where the lim means š‘ƒāˆ’lim and 1š›½(š‘—,š‘˜,š‘,š‘ž)=(š‘+1)(š‘ž+1)š‘ī“š‘žš‘—=0ī“š‘˜=0š‘Žš‘šš‘›š‘—š‘˜.(2.7)

Proof. Sufficiency. Suppose that the conditions (2.2)-(2.6) hold and š‘„=[š‘„š‘—š‘˜]āˆˆš‘āˆž2 with š‘ƒāˆ’limš‘—,š‘˜š‘„š‘—š‘˜=šæ, say. So that for every šœ–>0 there exists š‘>0 such that |š‘„š‘—š‘˜|<|ā„“|+šœ– whenever š‘—,š‘˜>š‘.
Then, we can write āˆžī“āˆžš‘—=0ī“š‘˜=0š›½(š‘—,š‘˜,š‘,š‘ž)š‘„š‘—š‘˜=š‘ī“š‘š‘—=0ī“š‘˜=0š›½(š‘—,š‘˜,š‘,š‘ž)š‘„š‘—š‘˜+āˆžī“š‘—=š‘š‘āˆ’1ī“š‘˜=0š›½(š‘—,š‘˜,š‘,š‘ž)š‘„š‘—š‘˜+š‘āˆ’1ī“āˆžš‘—=0ī“š‘˜=š‘š›½(š‘—,š‘˜,š‘,š‘ž)š‘„š‘—š‘˜+āˆžī“āˆžš‘—=š‘+1ī“š‘˜=š‘+1š›½(š‘—,š‘˜,š‘,š‘ž)š‘„š‘—š‘˜.(2.8) Therefore, |||||āˆžī“āˆžš‘—=0ī“š‘˜=0š›½(š‘—,š‘˜,š‘,š‘ž)š‘„š‘—š‘˜|||||ā‰¤ā€–š‘„ā€–š‘ī“š‘š‘—=0ī“š‘˜=0||||š›½(š‘—,š‘˜,š‘,š‘ž)+ā€–š‘„ā€–āˆžī“š‘—=š‘š‘āˆ’1ī“š‘˜=0||š›½(š‘—,š‘˜,š‘,š‘ž)š‘„š‘—š‘˜||+ā€–š‘„ā€–š‘āˆ’1ī“āˆžš‘—=0ī“š‘˜=š‘||||+ī€·||šæ||ī€ø|||||š›½(š‘—,š‘˜,š‘,š‘ž)+šœ–āˆžī“āˆžš‘—=š‘+1ī“š‘˜=š‘+1|||||.š›½(š‘—,š‘˜,š‘,š‘ž)(2.9) Letting š‘,š‘žā†’āˆž and using the conditions (2.2)ā€“(2.6), we get |||||limāˆžš‘,š‘žā†’āˆžī“āˆžš‘—=0ī“š‘˜=0š›½(š‘—,š‘˜,š‘,š‘ž)š‘„š‘—š‘˜|||||ā‰¤ī€·||šæ||ī€ø+šœ–š›¼.(2.10) Since šœ– is arbitrary, š¶1āˆ’limš“š‘„=š›¼šæ. Hence š“āˆˆ(š‘āˆž2,š¶1)š›¼, that is, š“ is š¶1-multiplicative.

Necessity. Suppose that A is š¶1-multiplicative. Then, by the definition, the A-transform of š‘„ exists and š“š‘„āˆˆš¶1 for each š‘„āˆˆš‘āˆž2. Therefore, š“š‘„ is also bounded. Then, we can write supāˆžš‘š,š‘›ī“āˆžš‘—=0ī“š‘˜=0||š‘Žš‘šš‘›š‘—š‘˜š‘„š‘—š‘˜||<š‘€<āˆž,(2.11) for each š‘„āˆˆš‘āˆž2. Now, let us define a sequence š‘¦=[š‘¦š‘—š‘˜] by š‘¦š‘—š‘˜=īƒÆsgnš‘Žš‘šš‘›š‘—š‘˜,0ā‰¤š‘—ā‰¤š‘Ÿ,0ā‰¤š‘˜ā‰¤š‘Ÿ,0,otherwise,(2.12) ā€‰š‘š,š‘›=0,1,2,ā€¦. Then, the necessity of (10) follows by considering the sequence š‘¦=[š‘¦š‘—š‘˜] in (2.11).
Also, by the assumption, we have limāˆžš‘,š‘žā†’āˆžī“āˆžš‘—=0ī“š‘˜=0š›½(š‘—,š‘˜,š‘,š‘ž)š‘„š‘—š‘˜=š›¼limš‘—,š‘˜ā†’āˆžš‘„š‘—š‘˜.(2.13) Now let us define the sequence š‘’š‘–š‘™ as follows: š‘’š‘–š‘™=īƒÆ1,(š‘—,š‘˜)=(š‘–,š‘™),0,otherwise,(2.14) and write š‘ š‘™=āˆ‘š‘–š‘’š‘–š‘™(š‘™āˆˆā„•), š‘Ÿš‘–=āˆ‘š‘™š‘’š‘–š‘™(š‘–āˆˆā„•). Then, the necessity of (2.2), (2.4), and (2.5) follows from š¶1āˆ’limš“š‘’š‘–š‘™, š¶1āˆ’limš“š‘Ÿš‘— and š¶1āˆ’limš“š‘ š‘˜, respectively.

Note that when š›¼=1, the above theorem gives the characterization of š“āˆˆ(š‘āˆž2,š¶1)reg. Now, we are ready to construct our main theorem.

Theorem 2.2. For every bounded double sequence x, š¶āˆ—1(š“š‘„)ā‰¤š›¼šæ(š‘„),(2.15) or (š‘ƒš¶āˆ’š‘š‘œš‘Ÿš‘’{š“š‘„}āŠ†š›¼(š‘ƒāˆ’š‘š‘œš‘Ÿš‘’{š‘„})) if and only if š“ is š¶1-multiplicative and limsupāˆžš‘,š‘žā†’āˆžī“āˆžš‘—=0ī“š‘˜=0||||š›½(š‘—,š‘˜,š‘,š‘ž)=š›¼.(2.16)

Proof. Necessity. Let (2.15) hold and for all š‘„āˆˆā„“2āˆž. So, since š‘āˆž2āŠ‚ā„“2āˆž, then, we get š›¼(āˆ’šæ(āˆ’š‘„))ā‰¤āˆ’š¶āˆ—1(āˆ’š“š‘„)ā‰¤š¶āˆ—1(š“š‘„)ā‰¤š›¼šæ(š‘„).(2.17) That is, š›¼liminfš‘„ā‰¤āˆ’š¶āˆ—1(āˆ’š“š‘„)ā‰¤š¶āˆ—1(š“š‘„)ā‰¤š›¼limsupš‘„,(2.18) where āˆ’š¶āˆ—1(āˆ’š“š‘„)=liminfāˆžš‘,š‘žā†’āˆžī“āˆžš‘—=0ī“š‘˜=0š›½(š‘—,š‘˜,š‘,š‘ž)š‘„š‘—š‘˜.(2.19) By choosing š‘„=[š‘„š‘—š‘˜]āˆˆš‘2āˆž, we get from (2.17) that āˆ’š¶āˆ—1(āˆ’š“š‘„)=š¶āˆ—1(š“š‘„)=š¶1āˆ’limš“š‘„=š›¼limš‘„.(2.20) This means that š“ is š¶1-multiplicative.
By Lemmaā€‰ā€‰3.1 of Patterson [9], there exists a š‘¦āˆˆā„“2āˆž with ||š‘¦||ā‰¤1 such that š¶āˆ—1(š“š‘¦)=limsupāˆžš‘,š‘žā†’āˆžī“āˆžš‘—=0ī“š‘˜=0š›½(š‘—,š‘˜,š‘,š‘ž).(2.21) If we choose š‘¦=š‘£=[š‘£š‘—š‘˜], it follows š‘£š‘—š‘˜=īƒÆ1ifš‘—=š‘˜,0,elsewhere.(2.22) Since ā€–š‘£š‘—š‘˜ā€–ā‰¤1, we have from (2.15) that š›¼=š¶āˆ—1(š“š‘£)=limsupāˆžš‘,š‘žā†’āˆžī“āˆžš‘—=0ī“š‘˜=0||||ī€·š‘£š›½(š‘—,š‘˜,š‘,š‘ž)ā‰¤š›¼šæš‘—š‘˜ī€øā‰¤š›¼ā€–š‘£ā€–ā‰¤š›¼.(2.23) This gives the necessity of (2.16).

Sufficiency. Suppose that š“ is š¶1-regular and (2.16) holds. Let š‘„=[š‘„š‘—š‘˜] be an arbitrary bounded sequence. Then, there exist š‘€,š‘>0 such that š‘„š‘—š‘˜ā‰¤š¾ for all š‘—,š‘˜ā‰„0. Now, we can write the following inequality: |||||āˆžī“āˆžš‘—=0ī“š‘˜=0š›½(š‘—,š‘˜,š‘,š‘ž)š‘„š‘—š‘˜|||||=|||||āˆžī“āˆžš‘—=0ī“š‘˜=0ī‚µ||š›½||(š‘—,š‘˜,š‘,š‘ž)+š›½(š‘—,š‘˜,š‘,š‘ž)2āˆ’||||š›½(š‘—,š‘˜,š‘,š‘ž)āˆ’š›½(š‘—,š‘˜,š‘,š‘ž)2ī‚¶š‘„š‘—š‘˜||||ā‰¤āˆžī“āˆžš‘—=0ī“š‘˜=0||š›½||||š‘„(š‘—,š‘˜,š‘,š‘ž)š‘—š‘˜||+āˆžī“āˆžš‘—=0ī“š‘˜=0||ī€·||š›½||ī€øš‘„(š‘—,š‘˜,š‘,š‘ž)āˆ’š›½(š‘—,š‘˜,š‘,š‘ž)š‘—š‘˜||ā‰¤ā€–š‘„ā€–š‘€ī“š‘š‘—=0ī“š‘˜=0||||š›½(š‘—,š‘˜,š‘,š‘ž)+ā€–š‘„ā€–āˆžī“š‘š‘—=š‘€+1ī“š‘˜=0||||š›½(š‘—,š‘˜,š‘,š‘ž)+ā€–š‘„ā€–š‘€ī“āˆžš‘—=0ī“š‘˜=š‘+1||||š›½(š‘—,š‘˜,š‘,š‘ž)+supš‘—,š‘˜ā‰„š‘€,š‘||š‘„š‘—š‘˜||āˆžī“āˆžš‘—=š‘€+1ī“š‘˜=š‘+1||||š›½(š‘—,š‘˜,š‘,š‘ž)+ā€–š‘„ā€–āˆžī“āˆžš‘—=0ī“š‘˜=0ī€·||||ī€ø.š›½(š‘—,š‘˜,š‘,š‘ž)āˆ’š›½(š‘—,š‘˜,š‘,š‘ž)(2.24) Using the condition of š¶1-multiplicative and condition (2.16), we get š¶āˆ—1(š“š‘„)ā‰¤š›¼šæ(š‘„).(2.25) This completes the proof of the theorem.

Theorem 2.3. For š‘„,š‘¦āˆˆā„“āˆž2, if š¶1āˆ’lim|š‘„āˆ’š‘¦|=0, then š¶1āˆ’core{š‘„}=š¶1āˆ’core{š‘¦}.

Proof. Since š¶2āˆ’lim|š‘„āˆ’š‘¦|=0, we have š¶1āˆ’lim(š‘„āˆ’š‘¦)=0 and š¶1āˆ’lim(āˆ’(š‘„āˆ’š‘¦))=0. Using definition of š¶1āˆ’core, we take š¶āˆ—1(š‘„āˆ’š‘¦)=āˆ’š¶āˆ—1(āˆ’(š‘„āˆ’š‘¦))=0. Since š¶āˆ—1 is sublinear, 0=āˆ’š¶āˆ—1(āˆ’(š‘„āˆ’š‘¦))ā‰¤āˆ’š¶āˆ—1(āˆ’š‘„)āˆ’š¶āˆ—1(š‘¦).(2.26) Therefore, š¶āˆ—1(š‘¦)ā‰¤āˆ’š¶āˆ—1(āˆ’š‘„). Since āˆ’š¶āˆ—1(āˆ’š‘„)ā‰¤š¶āˆ—1(š‘„), this implies that š¶āˆ—1(š‘¦)ā‰¤š¶āˆ—1(š‘„).By an argument similar as above, we can show that š¶āˆ—1(š‘„)ā‰¤š¶āˆ—1(š‘¦). This completes the proof.

Acknowledgment

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