Compact Operators for Almost Conservative and Strongly Conservative Matrices

Abstract

We obtain the necessary and sufficient conditions for an almost conservative matrix to define a compact operator. We also establish some necessary and sufficient (or only sufficient) conditions for operators to be compact for matrix classes , where . These results are achieved by applying the Hausdorff measure of noncompactness.

1. Introduction and Preliminaries

For some basic definitions and notations of this section we refer to [1, 2]. Let denote the space of all complex sequences , and let be the set of all sequences that terminate in zeros. Let , , and denote the spaces of all bounded, convergent, and null sequences, respectively. We will write and for the spaces of all convergent and absolutely convergent series, respectively. Further, we will use the conventions that and where at the th place for each .

For the sequence spaces and , we write which is called the multiplier space of and . The -, -, and -duals of a sequence space , which are respectively denoted by , , and , are 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 th 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 .

For arbitrary sequence spaces and , we write for the class of all infinite matrices that map into . Thus if and only if for all and for all .

The theory of spaces is the most powerful tool in the characterization of matrix transformations between sequence spaces.

A sequence space is called a space if it is a Banach space with continuous coordinates , where denotes the complex field and for all and every .

The sequence spaces , , and are spaces with the usual sup norm given by , where the supremum is taken over all . Also, the space is a space with the usual -norm defined by .

If is a space and , then we write provided the expression on the right exists and is finite which is the case whenever , where is the unit sphere in ; that is, .

A sequence in a linear metric space is called a Schauder basis (or briefly basis) for if for every there exists a unique sequence of scalars such that ; that is, , where is known as the -section of . The series which has the sum is called the expansion of , and is called the sequence of coefficients of with respect to the basis .

Let and be Banach spaces. Then, we write for the set of all bounded linear operators , which 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 . An operator is said to be of finite rank if , where denotes the range space of . An operator of finite rank is clearly compact. Further, we write for the class of all compact operators from to . Let us remark that every compact operator in is bounded; that is, . More precisely, the class is a closed subspace of the Banach space with the operator norm.

Finally, the following known results are fundamental for our investigation.

Lemma 1. Let denote any of the spaces , , or . Then, one has and for all .

Lemma 2. Let and be spaces. Then, one has ; that is, every matrix defines an operator by for all .

2. The Hausdorff Measure of Noncompactness

Most of the definitions, notations, and basic results of this section are taken from [3]. Throughout, we will write for the collection of all bounded subsets of a metric space . If , then the Hausdorff measure of noncompactness of the set , denoted by , is defined to be the infimum of the set of all reals such that can be covered by a finite number of balls of radii and centers in . This can equivalently be redefined as follows:

The function is called the Hausdorff measure of noncompactness.

If , , and are bounded subsets of a metric space , then we have 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 be the Hausdorff measures of noncompactness on and , respectively. An operator is said to be ,-bounded if for all and there exists a constant such that for all . If an operator is ,-bounded then the number for all is called the -measure of noncompactness of  . If , then we write .

Let and be Banach spaces and . Then, the Hausdorff measure of noncompactness of , denoted by , can be determined by and we have that Furthermore, the function is more applicable when is a Banach space. The most effective way in the characterization of compact operators between the Banach spaces is by applying the Hausdorff measure of noncompactness. The following result of Goldenštein et al. [4, Theorem 1] gives an estimate for the Hausdorff measure of noncompactness in Banach spaces with Schauder bases.

Lemma 3. Let be a Banach space with a Schauder basis and and the projector onto the linear span of . Then, one has where and the operator , defined for each by , is called the projector onto the linear span of . Besides, all operators and are equibounded, where denotes the identity operator on .

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

Lemma 4. Let be a bounded subset of the normed space , where is for or . If is the operator defined by for all , then one has It is easy to see that for Also, it is known that is a Schauder basis for the space and every sequence has a unique representation , where . Thus, one defines the projector , onto the linear span of , by for all with . In this situation, one has the following.

Lemma 5. Let and be the projector onto the linear span of . Then, one has where is the identity operator on .

3. Almost Conservative Matrices

A continuous linear functional on is said to be a Banach limit if it has the following properties: (i)   if , (ii)  , and (iii)  ; where is a shift operator defined by .

A bounded sequence is said to be almost convergent (Lorentz [5]) to the value if all of its Banach limits coincide; that is, for all Banach limits .

Lorentz established the following characterization.

A sequence is almost convergent to the number if and only if as uniformly in , where

The number is called the generalized limit of , and we write . We denote the set of all almost convergent sequences by ; that is,

Remark 6. Note that and each inclusion is proper.

Remark 7. Since , we have and hence . Therefore, it is natural by (4) and Lemma 1 that for all .

Remark 8 (see [6]). is a BK-space with .

Remark 9 (see [6]). is a nonseparable closed subspace of .
Using the idea of almost convergence, King [7] defined and characterized the almost conservative and almost regular matrices.
An infinite matrix is said to be almost conservative if for all , and we denote it by . If in addition , then is called almost regular.

Remark 10 (see [7, Theorem  1]). A matrix is almost conservative if and only if(i),(ii) for each ,(iii).
Now, we prove the following.

Theorem 11. Let be an almost conservative matrix. Then, one has

Proof. Let us remark that the expression on the right of (17) exists and is finite by Remark 10(i). We write , for short. Since , we have by Lemma 2 that . Thus, we obtain by (8) that
We define the operators by for all . Then, we have where is the identity operator on . Thus, it follows by the elementary properties of the function that for all . Further, we have for every that for all . Therefore, by using (3), (4), and Lemma 2, we derive that
Thus, we obtain that and hence This and (19) yield (17). Finally, we get (18) from (9) and (17).
This completes the proof.

It is worth mentioning that the condition in (18) is only a sufficient condition for the operator to be compact, where is an almost conservative matrix. More precisely, the following example will show that it is possible for to be compact while . Hence, in general, we have just “if” in (18) of Theorem 11.

Example 12. Define the matrix by and for . Then, we have for all and hence ; that is, is almost conservative. Also, it is obvious that is of finite rank and so is compact. On the other hand, we have and hence for all . This implies that .

4. Compact Operators for Strongly Conservative Matrices

An infinite matrix is said to be strongly conservative if for all , and we denote it by . If in addition , then is called strongly regular (cf. [5]).

In this final section, we establish some necessary and sufficient (or only sufficient) conditions for operators to be compact for matrix classes , where .

We may begin with the following lemmas which will be needed in the sequel.

Lemma 13. If the matrix is in any of the classes , , or , then

Proof. This can be seen from the class characterized by Lorentz [5] and by using the fact that , , and .
This completes the proof of the theorem.

Lemma 14. If , then one has

Proof. It is trivial that (26) holds, since for all . Further, by combining (26) and Lemma 13, we have for every that which implies that (27) holds. Finally, it follows by (27) and Lemma 13 that (28) holds.
This completes the proof of the theorem.

Now, we prove the following result on the Hausdorff measure of noncompactness.

Theorem 15. Let be an infinite matrix. Then, one has the following.(i)If , then (ii)If , then where for all .(iii)If , then

Proof. Let us remark that the expressions in (29), (30), and (31) exist by Lemmas 13 and 14.
We write . Then, we obtain by (8) and Lemma 2 that
For (i), we have . Thus, it follows by applying Lemma 3 that where is the operator defined by for all . This yields that for all and every . Therefore, by using (3), (4), and Remark 7, we have for every that
This and (33) imply that
Hence, we get (29) by (32).
To prove (ii), we have . Thus, we are going to apply Lemma 4 to get an estimate for the value of in (32). For this, let be the projectors defined by (13). Then, we have for every that and hence for all and every , where and is the identity operator on .
Now, by using (32), we obtain by applying Lemma 5 that
Further, since , we have by combining Lemmas 13 and 14 that and for all . Consequently, we derive from (36) that for all and every . Therefore, it follows by (4) that
Hence, from (37) we get (30).
For (iii), we have . Thus, we define by for all . Then, the proof can be achieved similarly as the proof of Theorem 11.
This completes the proof of the theorem.

Finally, we conclude our work by the following corollary.

Corollary 16. Let be an infinite matrix. Then, one has the following.(i)If , then (ii)If , then where for all .(iii)If , then

Proof. This result follows from Theorem 15 by using (9).

Remark 17. As in our Example 12, it can be shown similarly that the equivalence in (42) of Corollary 16 does not hold.

Remark 18. Since matrix mappings between BK spaces define bounded linear operators between these spaces which are Banach spaces, it is natural to use the Hausdorff measure of noncompactness to obtain necessary and sufficient conditions for matrix operators between BK spaces to be compact operators. This technique has recently been used by several authors in many research papers (cf. [8–14]). Further the technique of measures of noncompactness has also been used in solving the infinite system of differential equations in some sequence spaces (see [15–17]).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.