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Hausdorff measure of noncompactness in some sequence spaces of a triple band matrix
Journal of Inequalities and Applications volume 2013, Article number: 503 (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 β-duals of a subset X of w are defined by
Let μ and γ be two sequence spaces and be an infinite matrix of real or complex numbers , where . Then we say that A defines a matrix mapping from μ into γ, and we denote it by writing if for every sequence , the sequence , the A-transform of x is in γ, where
The notation denotes the class of all matrices A such that . Thus, if and only if the series on the right-hand side of (1.1) converges for each and every , and we have for all . The matrix domain of an infinite matrix A in a sequence space μ is defined by
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
Let r, s and t be non-zero real numbers, and define the triple band matrix
Recently, Sömez  introduced the sequence spaces and as the matrix domain of the triangle in the spaces and , respectively. It obvious that and are BK-spaces with the same norm by
Throughout, for any sequence , we define the sequence which will be frequently used, as the -transform of a sequence , i.e.,
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.
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 X is a BK-space and . Let S and M be subsets of a metric space and . Then S is called an ε-net of M in X if for every there exists such that . Further the set S is finite, then the ε-net S of M is called a finite ε-net of M, and we say that M has a finite ε-net in X. A subset of a metric space is said to be totally bounded if it has a finite ε-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 Q, denoted by , is defined by
The function is called the Hausdorff measure of noncompactness [, p.387].
The basic properties of the Hausdorff measure of noncompactness can be found in [, Lemma 2]; for example, if and Q are bounded subsets of a metric space , then
Further, if X is a normed space, then the function χ has some additional properties connected with the linear structure, that is,
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]
Let T be a triangle. Then we have
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]
Let be a BK-space and Y be any of the spaces , c or . If , then we have
Lemma 3.4 [, Theorem 1.29]
Let X denote any of the spaces c, or . If and for all .
Lemma 3.5 Let X denote any of the spaces and . If , then we have and the equality
holds for every , where is the associated sequence defined by (2.2) and
Theorem 3.6 Let X denote any of the spaces or . Then we have
for all , where is as in Lemma 3.5.
Proof Let Y be the respective one of the spaces or , and take any . Then we have by Lemma 3.5 that and equality (3.2) holds for all sequences and which are connected by relation (2.2). Further, it follows by (2.1) that if and only if . Therefore, we derive from (3.1) and (3.2) that
and since , we obtain from Lemma 3.4 that
Lemma 3.7 Let X be any of the spaces or , let Y be the respective one of the spaces or , Z be a sequence space and be an infinite matrix. If , then such that for all sequences and which are connected by relation (2.2), where is the associated matrix defined by
Proof It can be similarly proved by the same technique as in [, Lemma 2.3]. □
Theorem 3.8 Let X be any of the spaces or , let be an infinite matrix and be the associated matrix. If A is any of the classes , or , then
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]
Let Q be a bounded subset of the normed space X, where X is for or . If () is an operator defined by for all , then we have
where I is the identity operator on X.
Further, we know by [, Theorem 1.10] that every has a unique representation , where . Thus, we define the projectors () by
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)]
Let and () be the projector onto the linear span of . Then we have
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]
Let X and Y be Banach spaces and . Then we have
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]
Let X denote any of the spaces or . If , then
Lemma 4.2 [, Theorem 3.7]
Let be a BK-space. Then we have
If , then
If , then
Theorem 4.3 Let X denote any of the spaces and . Then we have
If , then(4.1)
If , then
Proof Let . Since for all , we have from Lemma 3.6 that
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). □
Theorem 4.4 Let X denote any of the spaces or . If , then we have
Proof By combining Lemma 3.7 and Lemma 4.1, we deduce that the expression in (4.4) exists. We write for short. Then we obtain by (3.5) and Lemma 3.1 that
and , where is the class of all bounded subsets of c. Then we are going to apply Lemma 3.10 to get an estimate for the value of in (4.3). For this, let be the projectors defined by (3.4). Then we have for every that and hence
for all and every . Thus, from (4.6) and applying Lemma 3.10, we get that
Now, for every given , let be an associated sequence space defined by (2.2), where Y is the respective one of the spaces or . Since , we have by Lemma 3.7 that and . Further, it follows from Lemma 4.1 that the limits exist for all and . Thus we derive from (4.7) that
for . Furthermore, since if and only if , we obtain by (3.1) and Lemma 3.1
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]
Let be a BK-space. If , then
Lemma 4.6 [, Lemma 3.5]
Let . Then the inequalities
Theorem 4.7 Let X denote any of the spaces or . If , then we have
Proof Since , the sequence of nonnegative reals is nonincreasing and bounded by Lemma 4.5. Thus, the limit in (4.9) exists.
Now, let . Then we have by Lemma 3.2(a) that . Hence, it follows from (3.5) and Lemma 3.9 that
Since, , we obtain by Lemma 4.6 that
for all and every . On the other hand, since for all , we derive from (3.1) and Lemma 3.4
for all (). This, together with (4.11), implies that
for every (). Thus, we get (4.9) by passing to the limits in (4.12) as and using (4.10). This completes the proof. □
Theorem 4.8 Let X denote any of the spaces or . Then we have
If , then(4.13)
If , then(4.15)
If , then(4.17)
where and .
Proof Let be an infinite matrix and be the summation matrix, and we define by
that is, , and hence
Further, let and be the associated matrices, respectively. Then it can be easily seen that
Furthermore, we define the sequence by
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. □
Boos J: Classical and Modern Methods in Summability. Oxford University Press, New York; 2000.
Sömez A: Some new sequence spaces derived by the domain of the triple band matrix. Comput. Math. Appl. 2011, 62: 641–650. 10.1016/j.camwa.2011.05.045
Mursaleen M, Noman AK: Applications of the Hausdorff measure of noncompactness in some sequence spaces of weighted means. Comput. Math. Appl. 2010, 60: 1245–1258. 10.1016/j.camwa.2010.06.005
Mursaleen M, Noman AK: Compactness of matrix operators on some new difference sequence spaces. Linear Algebra Appl. 2012, 436: 41–52. 10.1016/j.laa.2011.06.014
Malkowsky E, Rakocčević V: On matrix domains of triangles. Appl. Math. Comput. 2007, 189(2):1146–1163. 10.1016/j.amc.2006.12.024
Djolović I, Malkowsky E: A note on compact operators on matrix domains. J. Math. Anal. Appl. 2008, 340(1):291–303. 10.1016/j.jmaa.2007.08.021
Kara EE, Başarır M:On compact operators and some Euler -difference sequence spaces. J. Math. Anal. Appl. 2011, 379: 499–511. 10.1016/j.jmaa.2011.01.028
Kara EE, Başarır M: On the B -difference sequence space derived by generalized weighted mean and compact operators. J. Math. Anal. Appl. 2012, 391: 67–81. 10.1016/j.jmaa.2012.02.031
Malkowsky E, Rakocčević V, Živković S: Matrix transformations between the sequence spaces , , ( ) and certain BK spaces. Appl. Math. Comput. 2004, 147(2):377–396. 10.1016/S0096-3003(02)00674-4
Malkowsky E: Compact matrix operators between some BK -spaces. In Modern Methods of Analysis and Its Applications. Edited by: Mursaleen M. Anamaya Publ., New Delhi; 2010:86–120.
Malkowsky E, Rakocčević V: An introduction into the theory of sequence spaces and measures of noncompactness. Zb. Rad. - Mat. Inst. (Beogr.) 2000, 9(17):143–234.
Djolović I, Malkowsky E: Matrix transformations and compact operators on some new m th-order difference sequences. Appl. Math. Comput. 2008, 198(2):700–714. 10.1016/j.amc.2007.09.008
Djolović I, Malkowsky E:A note on Fredholm operators on . Appl. Math. Lett. 2009, 11: 1734–1739.
Mursaleen M, Noman AK: Compactness by the Hausdorff measure of noncompactness. Nonlinear Anal. 2010, 73: 2541–2557. 10.1016/j.na.2010.06.030
We thank the referees for their careful reading of the original manuscript and for the valuable comments.
The author declares that they have no competing interests.
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Karaisa, A. Hausdorff measure of noncompactness in some sequence spaces of a triple band matrix. J Inequal Appl 2013, 503 (2013). https://doi.org/10.1186/1029-242X-2013-503
- Hausdorff measure of noncompactness
- triple band matrix
- sequence space