Abstract

We determine the conditions for some matrix transformations from , where the sequence space , which is related to the spaces, was introduced by Sargent (1960). We also obtain estimates for the norms of the bounded linear operators defined by these matrix transformations and find conditions to obtain the corresponding subclasses of compact matrix operators by using the Hausdorff measure of noncompactness.

1. Introduction and Preliminaries

We shall write for the set of all complex sequences . Let , and denote the sets of all finite, bounded, convergent, and null sequences, respectively. We write for . By and , we denote the sequences such that for , and and . For any sequence , let be its -section. Moreover, we write and for the sets of sequences with bounded and convergent partial sums, respectively.

A sequence in a linear metric space is called Schauder basis if for every , there is a unique sequence of scalars such that . A sequence space with a linear topology is called a - if each of the maps defined by is continuous for all . A -space is called an - if is complete linear metric space; a - is a normed -space. An -space is said to have if every sequence has a unique representation , that is, as .

The spaces , and all have Schauder bases but the space has no Schauder basis. Among the other classical sequence spaces, the spaces and have .

Let be a normed space. Then the unit sphere and closed unit ball in are denoted by and . If is a -space and , then we define provided the expression on the right-hand side exists and is finite.

The -, -, and -duals of a subset of are, respectively, defined by

Throughout this paper, the matrices are infinite matrices of complex numbers. If is an infinite matrix with complex entries , then we write instead of . Also, we write for the sequence in the row of , that is, for every . In addition, if , then we define the -transform of as the sequence , where provided the series on the right converges for each .

Let and be subsets of and an infinite matrix. Then, we say that defines a matrix mapping from into , and we denote it by writing if exists and is in for all . By , we denote the class of all infinite matrices that map into . Thus if and only if for all and for all .

Lemma 1.1 (see[1]). Let denote any of the symbols , , or . Then, we have , and , where and .

Lemma 1.2 (see[1, 2]). Let be any of the spaces , , , or . Then, we have on , where denotes the natural norm on the dual space .

Lemma 1.3 (see[1, 2]). Let and be -spaces. Then, we have(a), that is, every matrix defines an operator by for all ;(b)if has , then , that is, for every operator there exists a matrix such that for all .

Furthermore, we have the following results on the operator norms.

Lemma 1.4 (see[2]). Let be a -space and any of the spaces , , or . If , then where denotes the operator norm for the matrix .

Sargent [3] defined the following sequence spaces.

Let denote the space whose elements are finite sets of distinct positive integers. Given any element of , we denote by the sequence such that for and otherwise. Further

that is, is the set of those whose support has cardinality at most , and we get

For , the following sequence spaces were introduced by Sargent [3] and further studied in [4] where and denotes the set of all sequences that are rearrangements of .

Remark 1.5 ([3]). (i) The spaces and are spaces with their respective norms. (ii) If for all , then if for all , then , . (iii) for all of .(iv) and , where is any of the symbols , , or .

Recently, Makowsky and Mursaleen [5] have characterized the classes of compact operators on some -spaces, namely, ,  ,, and . In this paper, we determine the conditions for the classes of matrix transformations ,, and , and establish estimates for the norms of the bounded linear operators defined by these matrix transformations. Further, we obtain the necessary and sufficient (or only sufficient) conditions for the corresponding subclasses of compact matrix operators ,, and by using the Hausdorff measure of noncompactness.

2. The Hausdorff Measure of Noncompactness

Let be a normed space. Then the unit sphere and closed unit ball in are denoted by and . If and are Banach spaces then is the set of all bounded linear operators is a Banach space with the operator norm given by for all . A linear operator is said to be compact if the domain of is all of , and for every bounded sequence in , the sequence has a subsequence which converges in . We denote the class of all compact operators in by . An operator is said to be of finite rank if , where is the range space of . An operator of finite rank is clearly compact. In particular, if then we write for the set of all continuous linear functionals on with the norm .

The Hausdorff measure of noncompactness was defined by Goldenštein et al. in 1957 [6].

Let and be subsets of a metric space and . Then, is called an -net of in if for every there exists such that . Further, if the set is finite, then the -net of is called a finite -net of , and we say that has a finite -net in . A subset of a metric space is said to be totally bounded if it has a finite -net for every .

By , we denote the collection of all bounded subsets of a metric space . If , then the Hausdorff measure of noncompactness of the set , denoted by , is defined by

The function is called the Hausdorff measure of noncompactness.

The basic properties of the Hausdorff measure of noncompactness can be found in [2, 7–9] and for recent developments, see [10–18]. If , , and are bounded subsets of a metric space , then

Further, if is a normed space, then the function has some additional properties connected with the linear structure, for example,

Let and be Banach spaces and and the Hausdorff measures of noncompactness on and , respectively. An operator is said to be ,-bounded if for all and there exist a constant such that for all . If an operator is ,-bounded, then the number is called the -measure of noncompactness . If , then we write .

The most effective way in the characterization of compact operators between the Banach spaces is by applying the Hausdorff measure of noncompactness. This can be achieved as follows: Let and be Banach spaces and . Then, the Hausdorff measure of noncompactness of , denoted by , can be determined by and we have that is compact if and only if

Now, the following result gives an estimate for the Hausdorff measure of noncompactness in Banach spaces with Schauder bases. It is known that if is a Schauder basis for a Banach space , then every element has a unique representation , where are called the basis functionals. Moreover, for each , the operator defined by is called the projector onto the linear span of . Besides, all operators and are equibounded, where denotes the identity operator on .

Theorem 2.1 (see[7]). Let be a Banach space with a Schauder basis , , and the projector onto the linear span of . Then, we have where .

In particular, the following result shows how to compute the Hausdorff measure of noncompactness in the spaces and , which are -spaces with .

Theorem 2.2 (see[7]). Let be a bounded subset of the normed space , where is for or . If is the operator defined by for all , then we have

The Haudorff measure of noncompactness for has recently been determined in [19] as follows.

Theorem 2.3. Let be a bounded subset of . Then

3. Main Results

First we prove the following basic lemma.

Lemma 3.1. If , then the following hold
Proof. We write , for short. Since , we have   Further, since and hence for all . Consequently, the limits in (3.1) exist for all .  Now, let be given. Then there is a positive constant such that for all . Thus we have for all . Hence, we obtain from (3.1) that   This implies that , and since was arbitrary, we deduce that . But and hence (3.2) holds. Moreover, since is a space, (3.2) implies by (Wilansky [19, Theorem  7.2.9]). Therefore, we get (3.3) from (15) by using (1.3).  Now, define the matrix by for all . Then, it is obvious that for all . Also, it follows by (3.3) that Furthermore, we have from (3.1) that that is, for all . This leads us to the consequence that by (Malkowsky-Rakocevic [2, Theorem  1.23(c)]). Hence, for all , which yields (3.4).

This completes the proof of the lemma.

Theorem 3.2. (a) If , then (b) If , then where with for all .
(c) If , then Proof. We write , for short. Then, we have by (2.4) and Lemma 1.3 (a) that For (a), we have . Thus, it follows by Theorem 2.2 that where is the operator defined by for all . This yields that for all and every . Thus, by combining (1.1) and (1.3), we have for every that
Hence, by (3.13) we get (3.9).
To prove , we have . Thus, we are going to apply Theorem 2.1 to get an estimate for the value of in (3.12). For this, we know that every has a unique representation , where . Thus, we define the projectors by and for . Then, we have for every that and hence
for all and every . Obviously , hence for all . Further, for each , we define the sequence by and for . Then and . Therefore, by (3.15). Consequently, we have for all . Hence, from (3.12) we obtain by applying Theorem 2.1 that where Now, it is given that . Thus, it follows from Lemma 3.1 that the limits exist for all , and for all . Therefore, we derive from (3.15) that
for all and every . Consequently, we obtain by (1.3) that
Hence, we get (3.10) from (3.16).
Finally, to prove we define as in the proof of part (a) for all .   Then, it is clear that
Thus, it follows by the elementary properties of the function that for all . This and (3.12) together imply (3.11). This completes the proof of the theorem.