Matrix transformations and compact operators on some new mth-order difference sequences

Djolović, Ivana; Malkowsky, Eberhard

doi:10.1016/j.amc.2007.09.008

Cited by 26 publications

(19 citation statements)

References 8 publications

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“…Theorem 1.7 ([7,Lemma 5.5]). Let Q be a bounded subset of the normed space X , where X is p for 1 ≤ p < ∞ or c 0 .…”

Section: The Hausdorff Measure Of Noncompactnessmentioning

confidence: 93%

Applications of the Hausdorff measure of noncompactness in some sequence spaces of weighted means

Mursaleen¹,

Noman²

2010

Computers & Mathematics with Applications

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a b s t r a c tIn the present paper, we establish some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain operators on some sequence spaces of weighted means. Furthermore, by using the Hausdorff measure of noncompactness, we apply our results to characterize some classes of compact operators on those spaces. Background, notation and preliminariesIt seems to be quite natural, in view of the fact that matrix operators between BK -spaces are continuous, to find necessary and sufficient conditions for the entries of an infinite matrix to define a compact operator between such spaces. This can be achieved in many cases by applying the Hausdorff measure of noncompactness. In this section, we give some related definitions, notation and preliminary results. Compact operators and matrix transformationsLet X be a normed space. Then, we write S X for the unit sphere in X , that is, S X = {x ∈ X : x = 1}. If X and Y are Banach spaces, then B(X, Y ) denotes the set of all bounded (continuous) linear operators L : X → Y , which is a Banach space with the operator norm given by L = sup x∈S X L(x) Y for all L ∈ B(X, Y ). A linear operator L : X → Y is said to be compact if the domain of L is all of X and for every bounded sequence (x n ) in X , the sequence (L(x n )) has a subsequence which converges in Y . We write C(X, Y ) for the class of all compact operators in B(X, Y ). An operator L ∈ B(X, Y ) is said to be of finite rank if dim R(L) < ∞, where R(L) is the range space of L. An operator of finite rank is clearly compact.By w, we shall denote the space of all complex sequences. If x ∈ w, then we writewe write φ for the set of all finite sequences that terminate in zeros. Further, we use the conventions that e = (1, 1, . . .) and e (k) is the sequence whose only non-zero term is 1 in the kth place for each k ∈ N, where N = {0, 1, 2, . . .}. Any vector subspace of w is called a sequence space. We shall write ∞ , c and c 0 for the sequence spaces of all bounded, convergent and null sequences, respectively. Further, by 1 and p (1 < p < ∞), we denote the sequence spaces of all absolutely and p-absolutely convergent series, respectively. Moreover, we write bs, cs and cs 0 for the sequence spaces of all bounded, convergent and null series, respectively.

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“…Theorem 1.7 ([7,Lemma 5.5]). Let Q be a bounded subset of the normed space X , where X is p for 1 ≤ p < ∞ or c 0 .…”

Section: The Hausdorff Measure Of Noncompactnessmentioning

confidence: 93%

Applications of the Hausdorff measure of noncompactness in some sequence spaces of weighted means

Mursaleen¹,

Noman²

2010

Computers & Mathematics with Applications

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show abstract

“…If X has A K then we have B( X, Y ) ⊂ (X, Y ), that is, every L ∈ B( X, Y ) is given by a matrix A ∈ (X, Y ) such that Ax = L(x) for all x ∈ X. [11,Corollary 5.13].) If A ∈ (( ∞ ) T , c) or A ∈ ((c 0 ) T , c), then we have…”

Section: Lemma 63 Let X and Y Be B K -Spacesmentioning

confidence: 99%

“…In this section, we will show that L A is a compact operator for every matrix A ∈ (( ∞ ) T , c 0 ) or A ∈ (( ∞ ) T , c) by applying the Hausdorff measure of noncompactness and using some results in [11,13,14], where L A (x) = Ax for all x ∈ ( ∞ ) T . First we give some notations, definitions and well-known results.…”

Section: The Hausdorff Measure Of Noncompactnessmentioning

confidence: 99%

“…In this paper, we go on defining some new Euler sequence spaces by using generalized difference matrix order m and give the some topological properties of these spaces. Also, we characterize some compact matrix classes between these spaces by applying the Hausdorff measure of noncompactness and using some results in [11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%

“…(6.2) Theorem 6.5. (See[11, Lemma 5.5].) Let Q be a bounded subsets of the normed space X , where X is l p for 1 p < ∞ or c 0 .…”

mentioning

confidence: 99%

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On compact operators and some EulerB(m)-difference sequence spaces

Kara

Başarır

2011

Journal of Mathematical Analysis and Applications

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Keywords: B (m) -difference sequence spaces Schauder basis α-, β-and γ -duals Matrix transformations Compact operators Hausdorff measure of noncompactness Altay and Ba¸sar (2005) [1] and Altay, Ba¸sar and Mursaleen (2006) [2] introduced the Euler sequence spaces e t 0 , e t c and e t ∞ . Başarır and Kayıkçı (2009) [3] defined the B (m) -difference matrix and studied some topological and geometric properties of some generalized Riesz B (m) -difference sequence space. In this paper, we introduce the Euler B (m) -difference sequence spaces e t 0 (B (m) ), e t c (B (m) ) and e t ∞ (B (m) ) consisting of all sequences whose B (m) -transforms are in the Euler spaces e t 0 , e t c and e t ∞ , respectively. Moreover, we determine the α-, β-and γ -duals of these spaces and construct the Schauder basis of the spaces e t 0 (B (m) )and e t c (B (m) ). Finally, we characterize some matrix classes concerning the spaces e t 0 (B (m) ) and e t c (B (m) ) and give the characterization of some classes of compact operators on the sequence spaces e t 0 (B (m) ) and e t ∞ (B (m) ).

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A note on Fredholm operators on (c0)T

Djolović

Malkowsky

2009

Applied Mathematics Letters

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Matrix transformations and compact operators on some new mth-order difference sequences

Cited by 26 publications

References 8 publications

Applications of the Hausdorff measure of noncompactness in some sequence spaces of weighted means

Applications of the Hausdorff measure of noncompactness in some sequence spaces of weighted means

On compact operators and some EulerB(m)-difference sequence spaces

A note on Fredholm operators on (c0)T

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