Characterization of -Core and Absolute Equivalence of Double Sequences

Abstract

The -core of a double sequence has been defined and it is studied by many authors. In this paper, we have determined two permutations and on the set of natural numbers for which for all .

1. Introduction and Preliminaries

A double sequence is said to be convergent in the Pringsheim sense or P-convergent if for every there exists such that whenever [1]. In this case, we write . By , we mean the space of all P-convergent sequences.

A double sequence is said to be bounded if there exists a positive number such that for all , that is, if

(1.1)

By we will denote the set of all bounded double sequences. We note that in contrast to the case for single sequences, a convergent double sequence needs not to be bounded. So, by we will denote the space of all real bounded and convergent double sequences.

Let be a four-dimensional infinite matrix of real numbers for all The sums

(1.2)

are called the -transforms of the double sequence and we will denote it by . We say that a sequence is -summable to the limit if the -transform of exists for all and convergent to in the Pringsheim sense, that is,

(1.3)

Móricz and Rhoades [2] have defined almost convergence of a double sequence as follows.

A double sequence of real numbers is said to be almost convergent to a limit if

(1.4)

uniformly in . By we denote the set of all almost convergent double sequences.

Recall that Knopp's Core of a single bounded sequence is the closed interval in [3, page 138]. In the sense of Knopp's Core, -core of a double sequence was introduced by Patterson as the closed interval in [4], where the definitions of (Pringsheim limit inferior) and (Pringsheim limit superior) are given as follows. Let and . Then

(1.5)

After this definition, this concept has been studied by many authors. For example see in [5–11] and the others.

Let denote the set of all natural numbers. A bijective function is said to be a permutation. In this paper we have determined two permutations and for which -core -core for all .

A two-dimensional matrix transformation is said to be regular (see [3, page 64]) if it maps every convergent sequence into a convergent sequence with the same limit. In 1926, Robison presented a four-dimensional analogue of regularity for double sequences in which he added an additional assumption of boundedness. A four-dimensional matrix is said to be -regular if it maps every bounded P-convergent sequence into a P-convergent sequence with the same P-limit.

Lemma 1.1 (see [12, 13]).

The four-dimensional matrix is bounded-regular or -regular if and only if

(1.6)

Lemma (see [4]).

If is a four-dimensional matrix, then for all real-valued double sequences ,

(1.7)

if and only if is RH-regular and

(1.8)

Now let us state the definition given in [14] for absolutely equivalent -regular matrices.

Definition 1.3.

Two -regular matrices and are said to be absolutely equivalent for a given class of sequences whenever

(1.9)

This means that and have the same limit or neither nor have a limit but their difference goes to zero.

The following proposition, lemma, and theorem characterizing the relationship between absolutely equivalent matrices and the -core are given in [14].

Proposition 1.4.

A necessary and sufficient condition for the -regular matrices and to be absolutely equivalent for all bounded sequences is that

(1.10)

Lemma.

If two double sequences and are such that , then -core -core .

Theorem.

-core -core for all bounded sequences if and only if is -regular and is absolutely equivalent to a nonnegative matrix for all bounded sequences.

If , then the four-dimensional matrix is said to be strongly -regular. In [2] the characterization of strongly -regular has been given as follows.

Theorem.

A matrix is strongly -regular if and only if is -regular and

(1.11)

2. The Main Results

Theorem 2.1.

If is a strongly -regular matrix and are two permutations such that

(2.1)

then -core -core for all , where is a positive integer and

(2.2)

Proof.

In the light of Lemma 1.5, it is enough to show that .

Let and . Then, it is clear that for each . Now, for , we can write

(2.3)

On the other hand, since is strongly -regular, by an easy calculation it can be seen that

(2.4)

By the same way, one can also see that

(2.5)

Now, conditions (1.11) imply that This completes the proof.

Here, let us specialize the permutations and . Let and be a permutation on such that and

(2.6)

for (see in [15, 16]). Also, let choose the permutation such that for all . Then, we have the following theorem.

Theorem 2.2.

If is a strongly -regular nonnegative matrix and are two permutations as above, then -core -core for all .

Proof.

By the same technique used in Theorem 2.1, one can see that . So, Lemma 1.5 implies that -core -core . But, since is an -regular nonnegative matrix, -core -core for all . This step completes the proof.