The Cesáro Core of Double Sequences

Abstract

We have characterized a new type of core for double sequences, 𝑃_𝐶-core, and determined the necessary and sufficient conditions on a four-dimensional matrix 𝐴 to yield 𝑃_𝐶-core{𝐴𝑥}⊆𝛼(𝑃-core{𝑥}) for all ℓ^∞2.

1. Introduction

A double sequence 𝑥=[𝑥_𝑗𝑘]^{∞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 |𝑥_𝑗𝑘−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‖=sup_{𝑗,𝑘≥0}||𝑥_𝑗𝑘||<∞.(1.1) 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 𝑐^∞2 the space of double sequences which are bounded and convergent.

A double sequence 𝑥=[𝑥_𝑗𝑘] is said to converge regularly if it converges in Pringsheim's sense and, in addition, the following finite limits exist: lliimm_{𝑘→∞𝑗→∞}𝑥_𝑗𝑘=ℓ_𝑗,(𝑗=1,2,3,…),𝑥_𝑗𝑘=𝑡_𝑗,(𝑘=1,2,3,…).(1.2) Let 𝐴=[𝑎^{𝑚𝑛𝑗𝑘}]^{∞𝑗,𝑘=0} be a four-dimensional infinite matrix of real numbers for all 𝑚,𝑛=0,1,…. The sums 𝑦_𝑚𝑛=^∞_∞𝑗=0_𝑘=0𝑎^{𝑚𝑛𝑗𝑘}𝑥_𝑗𝑘(1.3) are called the 𝐴-transforms of the double sequence 𝑥=[𝑥_𝑗𝑘]. We say that a sequence 𝑥=[𝑥_𝑗𝑘] is 𝐴-summable to the limit ℓ if the 𝐴-transform of 𝑥=[𝑥_𝑗𝑘] exists for all 𝑚,𝑛=0,1,… and is convergent to ℓ in the Pringsheim sense, that is, lim_{𝑝𝑝,𝑞→∞}_𝑞𝑗=0_𝑘=0lim_{𝑚,𝑛→∞}𝑎^{𝑚𝑛𝑗𝑘}𝑥_𝑗𝑘=𝑦_𝑚𝑛,𝑦_𝑚𝑛=ℓ.(1.4)

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 lim_{𝑚,𝑛→∞}𝑎l^{𝑚𝑛𝑗𝑘}im=_{∞𝑚,𝑛→∞}0,(𝑗,𝑘=0,1,…),_∞𝑗=0_𝑘=0lim_{∞𝑚,𝑛→∞}𝑎^{𝑚𝑛𝑗𝑘}=1,_𝑗=0lim_{∞𝑚,𝑛→∞}||𝑎^{𝑚𝑛𝑗𝑘}||=0,(𝑘=0,1,…),_𝑘=0||𝑎^{𝑚𝑛𝑗𝑘}||=0,(𝑗=0,1,…),_∞∞∑_𝑗=0∑_𝑘=0||𝑎^{𝑚𝑛𝑗𝑘}||≤𝐶<∞(𝑚,𝑛=0,1,…).(1.5) A double sequence 𝑥=[𝑥_𝑗𝑘] is said to be almost convergent (see [3]) to a number 𝐿 if lim_{𝑝,𝑞→∞}sup_{𝑠,𝑡≥0}1𝑝𝑞^𝑝_𝑞𝑗=0_𝑘=0𝑥_{𝑠+𝑗,𝑡+𝑘}=𝐿.(1.6) 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: lim_{𝑝,𝑞→∞}sup_{𝑠,𝑡≥0}1𝑝𝑞^𝑝_𝑞𝑗=0_𝑘=0𝑥_{𝜎𝑗(𝑠),𝜎𝑘(𝑡)}=ℓ,(1.7) where 𝜎^𝑗(𝑠)=𝜎(𝜎^𝑗−1(𝑠)). In this case, we write 𝜎−lim𝑥=ℓ. By 𝑉^2𝜎, 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, 𝑐^∞2⊂𝑉^2𝜎⊂ℓ^∞2. In the case 𝜎(𝑖)=𝑖+1, 𝜎-convergence of double sequences reduces to the almost convergence. A matrix 𝐴=[𝑎^{𝑚𝑛𝑗𝑘}]^{∞𝑗,𝑘=0} is said to be 𝜎-regular if 𝐴𝑥∈𝑉^𝜎2 for 𝑥=[𝑥_𝑗𝑘]∈𝑐^∞2 with 𝜎−lim𝐴𝑥=lim𝑥, and we denote this by 𝐴∈(𝑐^∞2,𝑉^𝜎2)_reg, (see [5, 6]). Mursaleen and Mohiuddine defined and characterized 𝜎-conservative and 𝜎-coercive matrices for double sequences [6].

A double sequence 𝑥=[𝑥_𝑗𝑘] 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∈𝑇ℓ^∞2∶lim_{𝑝,𝑞→∞}𝑇_𝑝𝑞(𝑥)=𝐿;𝐿=𝐶₁−lim𝑥_𝑝𝑞,(𝑥)=1(𝑝+1)(𝑞+1)^𝑝_𝑞𝑗=1_𝑘=1𝑥^{𝑚𝑛𝑗𝑘}.(1.8) We shall denote by 𝐶₁ the space of Cesáro convergent (𝐶₁-convergent) double sequences.

A matrix 𝐴=(𝑎^{𝑚𝑛𝑗𝑘}) is said to be 𝐶₁-multiplicative if 𝐴𝑥∈𝐶₁ for 𝑥=[𝑥_𝑗𝑘]∈𝑐^∞2 with 𝐶₁−lim𝐴𝑥=𝛼lim𝑥, and in this case we write 𝐴∈(𝑐^∞2,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=(𝑥_𝑘) 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=(𝑎_𝑛𝑘) should be regular and lim_𝑛→∞∑^∞𝑘=0|𝑎_𝑛𝑘|=1. Patterson [9] extended this idea for double sequences by defining the Pringsheim core (or P-core) of a real bounded double sequence 𝑥=[𝑥_𝑗𝑘] 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)=limsup_{𝑝,𝑞→∞}sup_𝑠,𝑡1𝑝𝑞^𝑝_𝑞𝑗=0_𝑘=0𝐶_𝜎𝑥_{𝑗+𝑠,𝑘+𝑡},(𝑥)=limsup_{𝑝,𝑞→∞}sup_𝑠,𝑡1𝑝𝑞^𝑝_𝑞𝑗=0_𝑘=0𝑥_{𝜎𝑗(𝑠),𝜎𝑘(𝑡)}.(1.9)

Then, MR- (Moricz-Rhoades) and 𝜎-core of a double sequence have been introduced by the closed intervals [−𝐿^∗(−𝑥),𝐿^∗(−𝑥)] and [−𝐶_𝜎(−𝑥),𝐶_𝜎(𝑥)], and also the inequalities 𝐿(𝐴𝑥)≤𝐿^∗(𝑥),𝐿^∗(𝐴𝑥)≤𝐿(𝑥),𝐿^∗(𝐴𝑥)≤𝐿^∗(𝑥),𝐿(𝐴𝑥)≤𝐶_𝜎(𝑥),𝐶_𝜎(𝐴𝑥)≤𝐿(𝑥)(1.10) 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=(𝑎^{𝑚𝑛𝑗𝑘}) to belong to the class (𝑐^∞2,𝐶₁)_𝛼. Further we investigate the necessary and sufficient conditions for the inequality 𝐶^∗1(𝐴𝑥)≤𝛼𝐿(𝑥)(1.11) for all 𝑥∈ℓ^2∞.

2. Main Results

Let us write 𝐶^∗1(𝑥)=limsup_{𝑝,𝑞→∞}1(𝑝+1)(𝑞+1)^𝑝_𝑞𝑗=0_𝑘=0𝑥_𝑗𝑘.(2.1) Then, we will define the 𝑃_𝐶-core of a realvalued bounded double sequence 𝑥=[𝑥_𝑗𝑘] by the closed interval [−𝐶^∗1(−𝑥),𝐶^∗1(𝑥)]. Since every bounded convergent double sequence is Cesáro convergent, we have 𝐶^∗1(𝑥)≤𝑃−limsup𝑥, and hence it follows that 𝑃_𝐶-core(𝑥)⊆𝑃-core(𝑥) for a bounded double sequence 𝑥=[𝑥_𝑗𝑘].

Lemma 2.1 .. A matrix 𝐴=(𝑎^{𝑚𝑛𝑗𝑘}) is 𝐶₁-multiplicative if and only if lim_{𝑝,𝑞→∞}𝛽li(m𝑗,_{∞𝑝,𝑞→∞}𝑘,𝑝,𝑞)=0(𝑗,𝑘=0,1,…),(2.2)_∞𝑗=0_𝑘=0lim_{∞𝑝,𝑞→∞}𝛽(𝑗,𝑘,𝑝,𝑞)=𝛼,(2.3)_𝑗=0lim_{∞𝑝,𝑞→∞}||𝛽(𝑗,𝑘,𝑝,𝑞)||=0(𝑘=0,1,…),(2.4)_𝑘=0||𝛽(𝑗,𝑘,𝑝,𝑞)||=0(𝑗=0,1,…),(2.5)_∞_∞𝑗=0_𝑘=0||𝑎^{𝑚𝑛𝑗𝑘}||≤𝐶<∞,(𝑚,𝑛=0,1,…),(2.6) where the lim means 𝑃−lim and 𝛽(𝑗,𝑘,𝑝,𝑞)=1(𝑝+1)(𝑞+1)^𝑝_𝑞𝑗=0_𝑘=0𝑎^{𝑚𝑛𝑗𝑘}.(2.7)

Proof. Sufficiency. Suppose that the conditions (2.2)-(2.6) hold and 𝑥=[𝑥_𝑗𝑘]∈𝑐^∞2 with 𝑃−lim_𝑗,𝑘𝑥_𝑗𝑘=𝐿, say. So that for every 𝜖>0 there exists 𝑁>0 such that |𝑥_𝑗𝑘|<|ℓ|+𝜖 whenever 𝑗,𝑘>𝑁.
Then, we can write ^∞_∞𝑗=0_𝑘=0𝛽(𝑗,𝑘,𝑝,𝑞)𝑥_𝑗𝑘=^𝑁_𝑁𝑗=0_𝑘=0𝛽(𝑗,𝑘,𝑝,𝑞)𝑥_𝑗𝑘+^∞_{𝑗=𝑁𝑁−1𝑘=0}𝛽(𝑗,𝑘,𝑝,𝑞)𝑥_𝑗𝑘+^𝑁−1_∞𝑗=0_𝑘=𝑁𝛽(𝑗,𝑘,𝑝,𝑞)𝑥_𝑗𝑘+^∞_∞𝑗_=𝑁+1_𝑘=𝑁+1𝛽(𝑗,𝑘,𝑝,𝑞)𝑥_𝑗𝑘.(2.8) Therefore, |||||_∞_∞𝑗=0_𝑘=0𝛽(𝑗,𝑘,𝑝,𝑞)𝑥_𝑗𝑘|||||≤‖𝑥‖_𝑁_𝑁𝑗=0_𝑘=0||𝛽(𝑗,𝑘,𝑝,𝑞)||+‖𝑥‖^∞_{𝑗=𝑁𝑁−1𝑘=0}+‖𝑥‖_𝑁−1||𝛽(𝑗,𝑘,𝑝,𝑞)𝑥_𝑗𝑘||_∞𝑗=0_𝑘=𝑁+||𝐿||+𝜖|||||_∞||𝛽(𝑗,𝑘,𝑝,𝑞)||_∞𝑗_=𝑁+1_𝑘=𝑁+1𝛽(𝑗,𝑘,𝑝,𝑞)|||||.(2.9) Letting 𝑝,𝑞→∞ and using the conditions (2.2)–(2.6), we get |||||lim_{∞𝑝,𝑞→∞}_∞𝑗=0_𝑘=0𝛽(𝑗,𝑘,𝑝,𝑞)𝑥_𝑗𝑘|||||≤||𝐿||+𝜖𝛼.(2.10) Since 𝜖 is arbitrary, 𝐶₁−lim𝐴𝑥=𝛼𝐿. Hence 𝐴∈(𝑐^∞2,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∈𝑐^∞2. Therefore, 𝐴𝑥 is also bounded. Then, we can write sup_∞𝑚,𝑛_∞𝑗=0_𝑘=0||𝑎^{𝑚𝑛𝑗𝑘}𝑥_𝑗𝑘||<𝑀<∞,(2.11) for each 𝑥∈𝑐^∞2. Now, let us define a sequence 𝑦=[𝑦_𝑗𝑘] by 𝑦_𝑗𝑘=sgn𝑎^{𝑚𝑛𝑗𝑘}0,,0ot≤her𝑗w≤is𝑟e,,0≤𝑘≤𝑟,(2.12)  𝑚,𝑛=0,1,2,…. Then, the necessity of (10) follows by considering the sequence 𝑦=[𝑦_𝑗𝑘] in (2.11).
Also, by the assumption, we have lim_{∞𝑝,𝑞→∞}_∞𝑗=0_𝑘=0𝛽(𝑗,𝑘,𝑝,𝑞)𝑥_𝑗𝑘=𝛼lim_{𝑗,𝑘→∞}𝑥_𝑗𝑘.(2.13) Now let us define the sequence 𝑒^𝑖𝑙 as follows: 𝑒^𝑖𝑙=10,,(o𝑗t,h𝑘e)rw=is(e𝑖,,𝑙),(2.14) and write 𝑠^𝑙=∑_𝑖𝑒^𝑖𝑙(𝑙∈ℕ), 𝑟^𝑖=∑_𝑙𝑒^𝑖𝑙(𝑖∈ℕ). Then, the necessity of (2.2), (2.4), and (2.5) follows from 𝐶₁−lim𝐴𝑒^𝑖𝑙, 𝐶₁−lim𝐴𝑟^𝑗 and 𝐶₁−lim𝐴𝑠^𝑘, respectively.

Note that when 𝛼=1, the above theorem gives the characterization of 𝐴∈(𝑐^∞2,𝐶₁)_reg. Now, we are ready to construct our main theorem.

Theorem 2.2. For every bounded double sequence x, 𝐶^∗1(𝐴𝑥)≤𝛼𝐿(𝑥),(2.15) or (𝑃_𝐶−𝑐𝑜𝑟𝑒{𝐴𝑥}⊆𝛼(𝑃−𝑐𝑜𝑟𝑒{𝑥})) if and only if 𝐴 is 𝐶₁-multiplicative and limsup_{∞𝑝,𝑞→∞}_∞𝑗=0_𝑘=0||𝛽(𝑗,𝑘,𝑝,𝑞)||=𝛼.(2.16)

Proof. Necessity. Let (2.15) hold and for all 𝑥∈ℓ^2∞. So, since 𝑐^∞2⊂ℓ^2∞, then, we get 𝛼(−𝐿(−𝑥))≤−𝐶^∗1(−𝐴𝑥)≤𝐶^∗1(𝐴𝑥)≤𝛼𝐿(𝑥).(2.17) That is, 𝛼liminf𝑥≤−𝐶^∗1(−𝐴𝑥)≤𝐶^∗1(𝐴𝑥)≤𝛼limsup𝑥,(2.18) where −𝐶^∗1(−𝐴𝑥)=liminf_{∞𝑝,𝑞→∞}_∞𝑗=0_𝑘=0𝛽(𝑗,𝑘,𝑝,𝑞)𝑥_𝑗𝑘.(2.19) By choosing 𝑥=[𝑥_𝑗𝑘]∈𝑐^2∞, we get from (2.17) that −𝐶^∗1(−𝐴𝑥)=𝐶^∗1(𝐴𝑥)=𝐶₁−lim𝐴𝑥=𝛼lim𝑥.(2.20) 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 𝐶^∗1(𝐴𝑦)=limsup_{∞𝑝,𝑞→∞}_∞𝑗=0_𝑘=0𝛽(𝑗,𝑘,𝑝,𝑞).(2.21) If we choose 𝑦=𝑣=[𝑣_𝑗𝑘], it follows 𝑣_𝑗𝑘=10,iefls𝑗ew=h𝑘e,re.(2.22) Since ‖𝑣_𝑗𝑘‖≤1, we have from (2.15) that 𝛼=𝐶^∗1(𝐴𝑣)=limsup_{∞𝑝,𝑞→∞}_∞𝑗=0_𝑘=0||𝛽(𝑗,𝑘,𝑝,𝑞)||≤𝛼𝐿𝑣_𝑗𝑘≤𝛼‖𝑣‖≤𝛼.(2.23) This gives the necessity of (2.16).

Sufficiency. Suppose that 𝐴 is 𝐶₁-regular and (2.16) holds. Let 𝑥=[𝑥_𝑗𝑘] be an arbitrary bounded sequence. Then, there exist 𝑀,𝑁>0 such that 𝑥_𝑗𝑘≤𝐾 for all 𝑗,𝑘≥0. Now, we can write the following inequality: |||||_∞_∞𝑗=0_𝑘=0𝛽(𝑗,𝑘,𝑝,𝑞)𝑥_𝑗𝑘|||||=|||||_∞_∞𝑗=0_𝑘=0−||𝛽||𝛽(𝑗(,𝑗𝑘,𝑘,𝑝,𝑝,𝑞,𝑞)||)2||+−𝛽𝛽(𝑗(,𝑗𝑘,𝑘,𝑝,𝑝,𝑞,𝑞))2𝑥^𝑗𝑘≤_∞||||_∞𝑗=0_𝑘=0+_∞||𝛽(𝑗,𝑘,𝑝,𝑞)||||𝑥_𝑗𝑘||_∞𝑗=0_𝑘=0≤‖𝑥‖_𝑀||||𝛽(𝑗,𝑘,𝑝,𝑞)||−𝛽(𝑗,𝑘,𝑝,𝑞)𝑥_𝑗𝑘||_𝑁𝑗=0_𝑘=0+‖𝑥‖_∞||𝛽(𝑗,𝑘,𝑝,𝑞)||_𝑁𝑗_=𝑀+1_𝑘=0+‖𝑥‖_𝑀||𝛽(𝑗,𝑘,𝑝,𝑞)||_∞𝑗=0_𝑘=𝑁+1+sup_{𝑗,𝑘≥𝑀,𝑁}||𝛽(𝑗,𝑘,𝑝,𝑞)||||𝑥_𝑗𝑘||^∞_∞𝑗_=𝑀+1_𝑘=𝑁+1+‖𝑥‖_∞||𝛽(𝑗,𝑘,𝑝,𝑞)||_∞𝑗=0_𝑘=0||𝛽(𝑗,𝑘,𝑝,𝑞)||−𝛽(𝑗,𝑘,𝑝,𝑞).(2.24) Using the condition of 𝐶₁-multiplicative and condition (2.16), we get 𝐶^∗1(𝐴𝑥)≤𝛼𝐿(𝑥).(2.25) This completes the proof of the theorem.

Theorem 2.3. For 𝑥,𝑦∈ℓ^∞2, 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 𝐶^∗1(𝑥−𝑦)=−𝐶^∗1(−(𝑥−𝑦))=0. Since 𝐶^∗1 is sublinear, 0=−𝐶^∗1(−(𝑥−𝑦))≤−𝐶^∗1(−𝑥)−𝐶^∗1(𝑦).(2.26) Therefore, 𝐶^∗1(𝑦)≤−𝐶^∗1(−𝑥). Since −𝐶^∗1(−𝑥)≤𝐶^∗1(𝑥), this implies that 𝐶^∗1(𝑦)≤𝐶^∗1(𝑥).By an argument similar as above, we can show that 𝐶^∗1(𝑥)≤𝐶^∗1(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.