Hausdorff measure of noncompactness in some sequence spaces of a triple band matrix
© Karaisa; licensee Springer. 2013
Received: 30 April 2013
Accepted: 18 September 2013
Published: 8 November 2013
The sequence spaces , have recently been introduced by Sömez (Comput. Math. Appl. 62:641-650, 2011). In this paper, we establish some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain matrix operators on the spaces , and by using the Hausdorff measure of noncompactness, we characterize some classes of compact operators on these spaces.
MSC:46A45, 40H05, 40C05.
By w, we shall denote the space of all real- or complex-valued sequences. Any vector subspace of w is called a sequence space. We shall write , c and for the spaces of all bounded, convergent and null sequences, respectively. Also, by and (), we denote the spaces of all absolutely and p-absolutely convergent series, respectively. Further, we shall write bs, cs for the spaces of all sequences associated with bounded and convergent series.
The theory of BK-spaces is the most powerful tool in the characterization of the matrix transformation between sequence spaces. A sequence space X is called a BK-space if it is a Banach space with the maps defined by being continuous for all , where ℂ denotes the complex field and .
The sequence spaces , c, and are BK-spaces with the usual sup-norm defined by and .
2 The sequence spaces and
Since the spaces and λ are norm isomorphic, one can easily observe that if and only if , where the sequences and are connected with relation (2.2); furthermore, , where λ is any of the sequences or .
3 Compactness by the Hausdorff measure of noncompactness
In the present paper, we establish some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain matrix operators on the spaces and . Further, by using the Hausdorff measure of noncompactness, we characterize some classes of compact operators on these spaces. It is quite natural to find condition for a matrix map between BK-spaces to define a compact operator since a matrix transformation between BK-spaces is continuous. This can be achieved by applying the Hausdorff measure of noncompactness. Recently, several authors characterized classes of compact operators given by infinite matrices on some sequence spaces by using this method. For example, in [3, 4], Mursaleen and Noman, Malkowsky and Rakoc̆ević , Djolović and Malkowsky  and Kara and Başarır [7, 8] established some identities or estimates for the operator norms and the Hausdorff measure of noncompactness of the linear operator given by infinite matrices that map an arbitrary BK-space or the matrix domain of triangles in an arbitrary BK-space. Further, they characterized some classes of compact operators on these spaces by using the Hausdorff measure of noncompactness. Now, we give some related definitions, notation and preliminary result.
Let X and Y be Banach spaces. Then we write for the set of all bounded (continuous) linear operators , which is a Banach space with the operator norm given by for all , where denotes the unit sphere in X, the sequence has a subsequence which converges in Y. By , we denote the class of all compact operators in . An operator is said to be of finite rank if , where denotes the range of L. An operator of finite rank is clearly compact.
The function is called the Hausdorff measure of noncompactness [, p.387].
We shall need the following known result for our investigation.
Lemma 3.1 [, Lemma 15(a)]
Let and Y be a BK-space. Then we also have , that is, every matrix defines an operator by for all .
Lemma 3.2 [, Theorem 3.8]
For arbitrary subsets X and Y of w, if and only if .
Further, if X and Y are BK-spaces and , then .
Lemma 3.3 [, Lemma 5.2]
Lemma 3.4 [, Theorem 1.29]
Let X denote any of the spaces c, or . If and for all .
for all , where is as in Lemma 3.5.
Proof It can be similarly proved by the same technique as in [, Lemma 2.3]. □
Proof This is immediate by combining Lemmas 3.2 and 3.6. □
The following result shows how to compute the Hausdorff measure of noncompactness in the BK-space .
Lemma 3.9 [, Theorem 3.3]
where I is the identity operator on X.
for all with , where and is the sequence whose only non-zero term is 1 in the nth place for each , where . In this situation, the following result gives an estimate for the Hausdorff measure of noncompactness in the BK-space c.
Lemma 3.10 [, Theorem 5(b)]
where I is the identity operator on c.
The next lemma is related to the Hausdorff measure of noncompactness of a bounded linear operator.
Lemma 3.11 [, Theorem 2.25]
4 Compact operators on the spaces and
In this subsection, we establish some identities or estimates for the Hausdorff measures of noncompactness of certain matrix operators on the spaces and . Further, we apply our results to characterize some classes of compact operators on those spaces. We begin with the following lemmas which will be used in proving our results.
Lemma 4.1 [, Lemma 3.1]
Lemma 4.2 [, Theorem 3.7]
- (a)If , then
- (b)If , then
- (a)If , then(4.1)and(4.2)
- (b)If , thenand
for all . Thus, we get (4.1) and (4.2) from (4.3) and Lemma 4.2(a). Part (b) can be proved similarly by using Lemma 4.2(b). □
for all . Thus, we get (4.4) and (4.5) from (4.8) and (3.6), respectively and this concludes the proof. □
Now, let ℱ denote the collection of all finite subsets of ℕ, and let () be the subcollection of ℱ consisting of all nonempty subsets of ℕ with elements that are grater than r.
Lemma 4.5 [, Proposition 4.3]
Lemma 4.6 [, Lemma 3.5]
Proof Since , the sequence of nonnegative reals is nonincreasing and bounded by Lemma 4.5. Thus, the limit in (4.9) exists.
for every (). Thus, we get (4.9) by passing to the limits in (4.12) as and using (4.10). This completes the proof. □
- (a)If , then(4.13)and(4.14)
- (b)If , then(4.15)and(4.16)
- (c)If , then(4.17)
where and .
provided the limits in (4.19) exist for all which is the case whenever .
Since , and , (4.13)-(4.18) are obtained from Theorems 4.3 and 4.4 by using Lemma 3.2. This completes the proof. □
We thank the referees for their careful reading of the original manuscript and for the valuable comments.
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