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Journal of Function Spaces
Volume 2014 (2014), Article ID 196489, 7 pages
http://dx.doi.org/10.1155/2014/196489
Research Article

Measure of Noncompactness for Compact Matrix Operators on Some BK Spaces

1Department of Mathematics, University of Giessen, Arndtstraβe 2, 35392 Giessen, Germany
2Department of Mathematics, Faculty of Arts and Sciences, Fatih University, Büyükçekmece, 34500 Istanbul, Turkey
3Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Received 20 May 2013; Accepted 31 October 2013; Published 10 February 2014

Academic Editor: Adrian Petrusel

Copyright © 2014 E. Malkowsky and A. Alotaibi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the spaces , , and of sequences that are strongly summable to 0, summable, and bounded with index by the Cesàro method of order 1 and establish the representations of the general bounded linear operators from the spaces into the spaces , , and . We also give estimates for the operator norm and the Hausdorff measure of noncompactness of such operators. Finally we apply our results to characterize the classes of compact bounded linear operators from and into and .

1. Introduction

The spaces , , and of all complex sequences that are strongly summable to zero, strongly summable, and strongly bounded, with index by the Cesàro method of order 1, were first introduced and studied by Maddox [1, 2]. Further recent studies on the spaces where the constant index in is replaced by the terms of a positive sequence can be found in [3, 4]. Extensive studies of generalizations of the spaces , , and to spaces of sequences of strong weighted means can be found in [5, 6].

In [7], a complete list was given of the characterizations of all matrix transformations from Maddox's spaces into the classical spaces , , , and of all bounded, convergent, and null sequences and of all absolutely convergent series. The characterizations of matrix transformations from the classical sequence spaces into Maddox's spaces with index were established in [8]. Furthermore, some classes of compact bounded linear operators between those spaces were characterized.

Recently, several authors applied the Hausdorff measure of noncompactness to characterize matrix transformations between sequence spaces that are matrix domains of triangles in the classical sequence spaces, for instance, in [914].

In this paper, we extend our studies from the normally considered matrix transformations to the general bounded linear operators from into , , and . We establish the representations of those operators, deduce estimates for their operator norms and Hausdorff measures of noncompactness, and characterize the corresponding classes of compact bounded linear operators.

2. Notations and Basic Results

In this section we list the notations, concepts, and basic results needed in the paper.

As usual, we denote by and the sets of all complex sequences and of all sequences that terminate in zeros; also let and for all be the sequences with for all and and for .

A Banach space is a BK space if each coordinate with for is continuous. A BK space is said to have AK if for every sequence .

Let be a normed space and and denote the unit sphere and closed unit ball in , respectively. If and are Banach spaces, then we write for the space of all bounded linear operators with the operator norm ; we write for the continuous dual of , that is, the space of all continuous linear functionals on with the norm . Furthermore, if is a normed sequence space, then we write for provided the expression on the right-hand side exists and is finite which is the case whenever is a BK space and [15, Theorem ].

For any subset of , the set is called the -dual of .

Let be an infinite matrix of complex numbers, let and be subsets of , and let. We write for the sequence in the th row of , , (provided all the series converge) and for the class of all matrices such that for all and for all . It is known that if and are BK spaces then every matrix defines an operator by for all [15, Theorem ] and if, in addition, has AK then every operator is given by a matrix such that for all [16, Theorem 1.9].

Throughout, let , and let be the conjugate number of ; that is, for and for . We write for the sets of all sequences that are strongly summable to , strongly summable, and strongly bounded, with index by the Cesàro method of order ; if , we write , , and , for short.

The following results are known and can be found in [1] and, for instance, in [7, Proposition 1.1].

For each sequence , the -limit for which is unique. We write for and .

The sets, , and are BK space with the equivalent block and sectional norms is a closed subspace of and is a closed subspace of ; has AK, and every sequence has a unique representation and has no Schauder basis. We always assume that , , and have the block norm, unless explicitly stated otherwise.

We put and . The following results are known and can be found, for instance, in [7, Proposition 2.1]: if and only if there are and a sequence such that Moreover

We need the following result where we assume that the initial and final spaces have the block and sectional norms, respectively.

Proposition 1. One has the following.(a)   if and only ifAlso .(b)   if and only if (11) holds and(c)    if and only if (11) holds and(d)   if and only if (11) and (12) hold and(e)   if and only if (11) and (13) hold and(f)If   in the cases above, then

Proof. (a) follows from [17, Corollary 1], (6), and (7).
(b) and (c) follow from [15, , p.   123], since and are closed subspaces of .
(d) and (e) follow from Parts (b) and (c) by [15, ].
(f) follows from [17, equation ()], (6), and (7).

3. Representation of Bounded Linear Operators

Here we establish the representations of the bounded linear operators in for and give estimates for the operator norms in each case. Throughout, we assume that and have the block and sectional norms, respectively.

We note that, since has AK, every is given by an infinite matrix , and its operator norm satisfies the inequalities in (16).

Theorem 2. (a) One has if and only if there exist a matrix and a sequence such that Moreover, one has
(b) One has if and only if there exist a matrix and a sequence with such that (17) holds; moreover, one has (18).
(c) One has if and only if there exist a matrix and a sequence with such that (17) holds; moreover, one has (18).

Proof. (a) First we assume and write for where denotes the th coordinate. Since is a space, it follows that for each , and hence we have by (9) This yields (17); moreover, we have by (10) It also follows from (21) and that and so . Furthermore, since for all , we have , and so and we obtain .
Now we show (18). We define for all and for each subset of by Then clearly , and we obtain by a well-known inequality (cf. [18]) and hence by the first inequality in (26) and by (10) for all and all , and so the first inequality in (18) follows. Also, we obtain for all from the second inequality in (26) and by (10) This implies the second inequality in (18).
Conversely, we assume that , that , and that (17) is satisfied. Let , and be given and let be the -limit of . Then there exists such that for all . Thus we have for all Since was arbitrary, we have We define the map with for all , where is the -limit of . Then trivially is linear, and it follows from (30) that and, since , we obtain . Furthermore, we have , and hence , and so, by (17), .
(b) First we assume . Then , and by Part (a) there are and such that (17) is satisfied; also clearly (18) is satisfied. It follows from (17), , and for each that there exist and for such that which is (19), and Now it follows from and (33) by Proposition 1(c) that .
Conversely, we assume that , that , and that (17) and (19) are satisfied. Then we have , and so by Part (a). Let be given and let be the -limit of . Then we have and, by (17), Since , the -limit of of exists and we have by (19) and (34) Therefore we have .
(c) The proof of Part (c) is similar to that of Part (b) with and .

Remark 3. It was shown in the proof of [19, Theorem 3.6] that if then with from (33) and in [19, equation (3.14)] that the -limit of for any sequence in is given by Let and and let be the -limit of ; then we obtain by (35) and (36) for the -limit of

4. Compact Operators

In this section, we establish estimates for the Hausdorff measures of noncompactness of linear operators and characterize some classes of compact operators from into , where and .

First we recall some useful definitions and results. The reader is referred to the monographs [2023] for the theory and applications of measures of noncompactness. Let and be Banach spaces and let be a linear operator. Then is said to be compact if its domain is all of and, for every bounded sequence in , the sequence has a convergent subsequence in . We denote the class of such operators by .

Let be a metric space, denote the open ball of radius and centre in , and denote the class of bounded subsets of . Then the map with is called the Hausdorff measure of noncompactness.

Let and be Banach spaces and let and be the Hausdorff measures of noncompactness on and . Then the operator is said to be -bounded if for every , and there exists a positive constant such that for every . If an operator is -bounded, then is called the -measure of noncompactness of . In particular, if , then we write .

The following useful results are well known.

Proposition 4. Let and be Banach spaces and . Then one has (see [23, Theorem 2.25]), (see [23, Corollary 2.26, equation (2.58)]).

We also need the following results which are an immediate consequence of [8, Proposition 3.2 and Lemma 3.5].

Proposition 5. (a) Let be the projectors onto the linear span of , the identity, and for . Then one has for all
(b) Let be the projectors onto the linear span of , the identity, and for . Then one has for all

Proof. (a) The inequalities in (42) follow from [8, Proposition 3.2 and Lemma 3.5].
(b) The identity in (42) follows from [8, Proposition 3.2 and Lemma 3.3(a)].

Now we give estimates for the Hausdorff measures of noncompactness of the general operators and . Let and . Then we write for any subset of the set .

Theorem 6. (a) Let . One uses the notations of Theorem 2 and writes for and for the matrix with for all and . Then one has
(b) Let . Then one has

Proof. (a) We assume .
Let be given, be the -limit of , and . Then we have from Theorem 2(b) and (17) that , where and , and it follows that Furthermore, the complex numbers and satisfying (19) and (33) exist by Theorem 2(b), and by Remark 3, so defined in (37) is the -limit of by Remark 3. Now let be the projector onto the linear span of and for , where is the identity on ; hence by (4). Let be given. We write for all and obtain for and, for , from (46) and (37) Since , we obtain by the same kind of argument as in the proof of (18) that Now the inequalities in (44) follow from (40) and (42).
(b) Now for , and as in the proof of Part (a) we obtain (48) with and replaced by and , respectively, and the inequalities in (45) follow from (40) and (43).

Corollary 7. (a) Let . Then is given by a matrix and one has, writing for the matrix with for all and , where is given by (33),
(b) Let . Then one has

Proof. (a) The estimates in (49) are easily obtained from [8, Theorem 3.6] with defined in (11).
(b) The estimates in (50) follow from those in (45) with for all .

We apply our results and close with the characterizations of the classes for and .

Corollary 8. Let . Then the necessary and sufficient conditions for can be read from Table 1 where

tab1
Table 1

Proof. The conditions in (1)–(4) are immediate consequences of (41) and the conditions in (50), (45), (49), and (44), respectively.

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.

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