Compact operators on some generalized mixed norm spaces

We establish identities or estimates for the Hausdorff measure of noncompactness of operators from some generalized mixed norm spaces into any of the spaces c0, c, ?1, and [?1, ??]<m(?)>. Furthermore we give necessary and sufficient conditions for the operators in these cases to be compact. Our results are complementary to those in [1, 3, 13].

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Compact Operators for Almost Conservative and Strongly Conservative Matrices

Abstract and Applied Analysis ◽

10.1155/2014/567317 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6 ◽

Cited By ~ 1

Author(s):

S. A. Mohiuddine ◽

M. Mursaleen ◽

A. Alotaibi

Keyword(s):

Compact Operator ◽

Hausdorff Measure ◽

Measure Of Noncompactness ◽

Sufficient Conditions ◽

Compact Operators ◽

Necessary And Sufficient Conditions ◽

Hausdorff Measure Of Noncompactness ◽

Matrix Classes ◽

Necessary And Sufficient ◽

Conservative Matrix

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(f,X), whereX=c,c0,l∞. These results are achieved by applying the Hausdorff measure of noncompactness.

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Measure of noncompactness of operators and matrices on the spacescandc0

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/ijmms/2006/46930 ◽

2006 ◽

Vol 2006 ◽

pp. 1-5 ◽

Cited By ~ 11

Author(s):

Bruno De Malafosse ◽

Eberhard Malkowsky ◽

Vladimir Rakocevic

Keyword(s):

Linear Operator ◽

Hausdorff Measure ◽

Measure Of Noncompactness ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Hausdorff Measure Of Noncompactness ◽

Necessary And Sufficient

In this note, using the Hausdorff measure of noncompactness, necessary and sufficient conditions are formulated for a linear operator and matrices between the spacescandc0to be compact. Among other things, some results of Cohen and Dunford are recovered.

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Hausdorff measure of noncompactness of matrix mappings on Cesàro spaces

Boletim da Sociedade Paranaense de Matemática ◽

10.5269/bspm.40448 ◽

2021 ◽

Vol 39 (1) ◽

pp. 157-167

Author(s):

G. Canan Hazar Güleç ◽

M. Ali Sarıgöl

Keyword(s):

Hausdorff Measure ◽

Measure Of Noncompactness ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Hausdorff Measure Of Noncompactness ◽

Cesàro Spaces ◽

Operator Norms ◽

Necessary And Sufficient ◽

Matrix Mappings

In this study we establish some identities or estimates for operator norms and the Hausdorff measure of noncompactness of certain operators on spaces |C_{α}|_{k}, which have more recently been introduced in [14]. Further, by applying the Hausdorff measure of noncompactness, we establish the necessary and sufficient conditions for such operators to be compact and so the some well known results are generalized.

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Matrix transformations between the sequence space BVP and certain BK spaces

Bulletin Classe des sciences mathematiques et natturalles ◽

10.2298/bmat0227033m ◽

2002 ◽

Vol 123 (27) ◽

pp. 33-46 ◽

Cited By ~ 11

Author(s):

Eberhard Malkowsky ◽

Vladimir Rakocevic ◽

Snezana Zivkovic-Zlatanovic

Keyword(s):

Linear Operator ◽

Hausdorff Measure ◽

Sequence Space ◽

Measure Of Noncompactness ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Matrix Transformations ◽

Hausdorff Measure Of Noncompactness ◽

Necessary And Sufficient ◽

Bk Spaces

In this paper, we characterize matrix transformations between the sequence space bvp (1 < p < ?) and certain BK spaces. Further?more, we apply the Hausdorff measure of noncompactness to give necessary and sufficient conditions for a linear operator between these spaces to be compact.

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Characterization of some classes of compact operators between certain matrix domains of triangles

Filomat ◽

10.2298/fil1605327d ◽

2016 ◽

Vol 30 (5) ◽

pp. 1327-1337

Author(s):

Ivana Djolovic ◽

Eberhard Malkowsky

Keyword(s):

Hausdorff Measure ◽

Measure Of Noncompactness ◽

Sufficient Conditions ◽

Compact Operators ◽

Fredholm Operators ◽

Hausdorff Measure Of Noncompactness ◽

Matrix Domains ◽

Matrix Operators

In this paper, we characterize the classes ((?1)T, (?1)?T ) and (cT, c?T) where T = (tnk)?n,k=0 and ?T=(?tnk)?n,k=0 are arbitrary triangles. We establish identities or estimates for the Hausdorff measure of noncompactness of operators given by matrices in the classes ((?1)T, (?1)?T ) and (cT, c?T). Furthermore we give sufficient conditions for such matrix operators to be Fredholm operators on (?1)T and cT. As an application of our results, we consider the class (bv, bv) and the corresponding classes of matrix operators. Our results are complementary to those in [2] and some of them are generalization for those in [3].

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Some notes on matrix mappings and their Hausdorff measure of noncompactness

Filomat ◽

10.2298/fil1405059m ◽

2014 ◽

Vol 28 (5) ◽

pp. 1059-1072 ◽

Cited By ~ 2

Author(s):

E. Malkowsky ◽

F. Özger ◽

A. Alotaibi

Keyword(s):

Hausdorff Measure ◽

Measure Of Noncompactness ◽

Sufficient Conditions ◽

Linear Operators ◽

Matrix Transformations ◽

Hausdorff Measure Of Noncompactness ◽

Bounded Linear ◽

Matrix Domains ◽

Necessary And Sufficient ◽

Matrix Mappings

We consider the sequence spaces s0?(?B), s(c)? (?B) and s?(?B) with their topological properties, and give the characterizations of the classes of matrix transformations from them into any of the spaces ?1, ?1, c0 and c. We also establish some estimates for the norms of bounded linear operators defined by those matrix transformations. Moreover, the Hausdorff measure of noncompactness is applied to give necessary and sufficient conditions for a linear operator on the sets s0?(?B), s(c)?(?B) and s?(?B) to be compact. We also close a gap in the proof of the characterizations by various authors of matrix transformations on matrix domains.

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Measures of noncompactness in (N¯qΔ)summable difference sequence spaces

Filomat ◽

10.2298/fil1815459m ◽

2018 ◽

Vol 32 (15) ◽

pp. 5459-5470

Author(s):

Ishfaq Malik ◽

Tanweer Jalal

Keyword(s):

Measure Of Noncompactness ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Linear Operators ◽

Sequence Spaces ◽

Difference Sequence ◽

Hausdorff Measure Of Noncompactness ◽

Measures Of Noncompactness ◽

Necessary And Sufficient ◽

Difference Sequence Spaces

In this paper we first introduce N?q?summable difference sequence spaces and prove some properties of these spaces. We then obtain the necessary and sufficient conditions for infinite matrices A to map these sequence spaces into the spaces c,c0, and l?. Finally, the Hausdorff measure of noncompactness is then used to obtain the necessary and sufficient conditions for the compactness of the linear operators defined on these spaces.

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Mixed-Norm Amalgam Spaces and Their Predual

Symmetry ◽

10.3390/sym14010074 ◽

2022 ◽

Vol 14 (1) ◽

pp. 74

Author(s):

Houkun Zhang ◽

Jiang Zhou

Keyword(s):

Duality Theory ◽

Fractional Integral ◽

Necessary Conditions ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Mixed Norm ◽

Fractional Integral Operators ◽

Primal Space ◽

Necessary And Sufficient ◽

Amalgam Spaces

In this paper, we introduce mixed-norm amalgam spaces (Lp→,Ls→)α(Rn) and prove the boundedness of maximal function. Then, the dilation argument obtains the necessary and sufficient conditions of fractional integral operators’ boundedness. Furthermore, the strong estimates of linear commutators [b,Iγ] generated by b∈BMO(Rn) and Iγ on mixed-norm amalgam spaces (Lp→,Ls→)α(Rn) are established as well. In order to obtain the necessary conditions of fractional integral commutators’ boundedness, we introduce mixed-norm Wiener amalgam spaces (Lp→,Ls→)(Rn). We obtain the necessary and sufficient conditions of fractional integral commutators’ boundedness by the duality theory. The necessary conditions of fractional integral commutators’ boundedness are a new result even for the classical amalgam spaces. By the equivalent norm and the operators Str(p)(f)(x), we study the duality theory of mixed-norm amalgam spaces, which makes our proof easier. In particular, note that predual of the primal space is not obtained and the predual of the equivalent space does not mean the predual of the primal space.

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The almost Dunford–Pettis property of semi-compact operators

Demonstratio Mathematica ◽

10.1515/dema-2013-0350 ◽

2012 ◽

Vol 45 (1) ◽

Author(s):

Belmesnaoui Aqzzouz ◽

Aziz Elbour

Keyword(s):

Compact Operator ◽

Sufficient Conditions ◽

Compact Operators ◽

Necessary And Sufficient Conditions ◽

Necessary And Sufficient

AbstractWe establish necessary and sufficient conditions for which each positive semi-compact operator (resp. the second power of a positive semi-compact operator) is almost Dunford–Pettis (resp. Dunford–Pettis).

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On strong summability and convergence

Filomat ◽

10.2298/fil1711095m ◽

2017 ◽

Vol 31 (11) ◽

pp. 3095-3123

Author(s):

Eberhard Malkowsky

Keyword(s):

Hausdorff Measure ◽

Measure Of Noncompactness ◽

Compact Operators ◽

Strong Summability ◽

Linear Operators ◽

Topological Properties ◽

Hausdorff Measure Of Noncompactness ◽

Bounded Sequences ◽

Convergent Sequences

We give a survey of the recent results concerning the fundamental topological properties of spaces of stronly summable and convergent sequences, their ?- and continuous duals, and the characterizations of classes of linear operators between them. Furthermore we demonstrate how the Hausdorff measure of noncompactness can be used in the characterization of classes of compact operators between the spaces of strongly summable and bounded sequences.

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