scholarly journals On strong summability and convergence

Filomat ◽  
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.

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

Filomat ◽  
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].

scholarly journals Some Compact Operators on the Hahn Space

2021 ◽  
Vol 1 (1) ◽  
pp. 1-15
Author(s):  
Eberhard Malkowsky
Keyword(s):  
Hausdorff Measure ◽  
Sequence Space ◽  
Measure Of Noncompactness ◽  
Compact Operators ◽  
Linear Operators ◽  
Hausdorff Measure Of Noncompactness ◽  
Real Numbers ◽  
Positive Real ◽  
Bounded Linear Operators ◽  
Bounded Linear

We establish the characterisations of the classes of bounded linear operators from the generalised Hahn sequence space $h_{d}$, where $d$ is an unbounded monotone increasing sequence of positive real numbers, into the spaces $[c_{0}]$, $[c]$ and $[c_{\infty}]$ of sequences that are strongly convergent to zero, strongly convergent and strongly bounded. Furthermore, we prove estimates for the Hausdorff measure of noncompactness of bounded linear operators from $h_{d}$ into $[c]$, and identities for the Hausdorff measure of noncompactness of bounded linear operators from $h_{d}$ to $[c_{0}]$, and use these results to characterise the classes of compact operators from $h_{d}$ to $[c]$ and $[c_{0}]$.

scholarly journals Generalised regularity and compactness of matrix operators

2020 ◽  
Vol 12 (2) ◽  
pp. 57-66
Author(s):  
E. Malkowsky
Keyword(s):  
Measure Of Noncompactness ◽  
Linear Operators ◽  
Matrix Transformations ◽  
Complex Numbers ◽  
Infinite Matrix ◽  
Regular Matrix ◽  
Hausdorff Measure Of Noncompactness ◽  
Complex Sequences ◽  
Matrix Operators ◽  
Convergent Sequences

A well-known result by Cohen and Dunford ([2], 1937) characterises the class of all regular compact linear operators. It follows that a regular matrix transformation cannot be compact. This means that if c denotes the set of all complex sequences of complex numbers, then an infinite matrix that maps c into c and preserves the limits cannot be compact. We obtained this result in a different way applying the theory of BK spaces from functional analysis and summability, and using the Hausdorff measure of noncompactness. Furthermore, we present the extension of this result to matrix transformations between the spaces c and the spaces of strongly summable sequences by the Cesaro method of order 1, and of strongly convergent sequences. We present new unified proofs for our main results.

scholarly journals Applications of Measure of Noncompactness in Matrix Operators on Some Sequence Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
M. Mursaleen ◽  
A. Latif
Keyword(s):  
Hausdorff Measure ◽  
Sequence Space ◽  
Measure Of Noncompactness ◽  
Linear Operators ◽  
Sequence Spaces ◽  
Matrix Transformations ◽  
Hausdorff Measure Of Noncompactness ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Matrix Operators

We determine the conditions for some matrix transformations fromn(ϕ), where the sequence spacen(ϕ), which is related to theℓpspaces, was introduced by Sargent (1960). We also obtain estimates for the norms of the bounded linear operators defined by these matrix transformations and find conditions to obtain the corresponding subclasses of compact matrix operators by using the Hausdorff measure of noncompactness.

scholarly journals Measure of Noncompactness for Compact Matrix Operators on Some BK Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
E. Malkowsky ◽  
A. Alotaibi
Keyword(s):  
Hausdorff Measure ◽  
Measure Of Noncompactness ◽  
Linear Operators ◽  
Operator Norm ◽  
Hausdorff Measure Of Noncompactness ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Matrix Operators ◽  
Bk Spaces ◽  
Cesàro Method

We study the spacesw0p,wp, andw∞pof sequences that are strongly summable to 0, summable, and bounded with indexp≥1by the Cesàro method of order 1 and establish the representations of the general bounded linear operators from the spaceswpinto the spacesw∞1,w1, andw01. 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 fromw0pandwpintow01andw1.

scholarly journals Compact Operators for Almost Conservative and Strongly Conservative Matrices

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
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.

scholarly journals Measure of noncompactness for compact matrix operators on some BK spaces

Filomat ◽  
2014 ◽  
Vol 28 (5) ◽  
pp. 1081-1086 ◽  
Author(s):  
A. Alotaibi ◽  
E. Malkowsky ◽  
M. Mursaleen
Keyword(s):  
Hausdorff Measure ◽  
Measure Of Noncompactness ◽  
Linear Operators ◽  
Matrix Transformations ◽  
Hausdorff Measure Of Noncompactness ◽  
Bounded Linear ◽  
Matrix Classes ◽  
The Matrix ◽  
Matrix Operators ◽  
Bk Spaces

In this paper, we characterize the matrix classes (?1, ??p )(1? p < 1). We also obtain estimates for the norms of the bounded linear operators LA defined by these matrix transformations and find conditions to obtain the corresponding subclasses of compact matrix operators by using the Hausdorff measure of noncompactness.

scholarly journals On composition operators of Fibonacci matrix and applications of Hausdorff measure of noncompactness

2022 ◽  
Vol 40 ◽  
pp. 1-24
Author(s):  
Bipan Hazarika ◽  
Anupam Das ◽  
Emrah Evren Kara ◽  
Feyzi Basar
Keyword(s):  
Hausdorff Measure ◽  
Measure Of Noncompactness ◽  
Composition Operators ◽  
Compact Operators ◽  
Hausdorff Measure Of Noncompactness ◽  
Infinite Matrices ◽  
Geometric Properties ◽  
Matrix Classes ◽  
Alpha Beta ◽  
Fibonacci Matrix

The aim of the paper is introduced the composition of the two infinite matrices $\Lambda=(\lambda_{nk})$ and $\widehat{F}=\left( f_{nk} \right).$ Further, we determine the $\alpha$-, $\beta$-, $\gamma$-duals of new spaces and also construct the basis for the space $\ell_{p}^{\lambda}(\widehat{F}).$ Additionally, we characterize some matrix classes on the spaces $\ell_{\infty}^{\lambda}(\widehat{F})$ and $\ell_{p}^{\lambda}(\widehat{F}).$ We also investigate some geometric properties concerning Banach-Saks type $p.$Finally we characterize the subclasses $\mathcal{K}(X:Y)$ of compact operators by applying the Hausdorff measure of noncompactness, where $X\in\{\ell_{\infty}^{\lambda}(\widehat{F}),\ell_{p}^{\lambda}(\widehat{F})\}$ and $Y\in\{c_{0},c, \ell_{\infty}, \ell_{1}, bv\},$ and $1\leq p<\infty.$

scholarly journals Some notes on matrix mappings and their Hausdorff measure of noncompactness

Filomat ◽  
2014 ◽  
Vol 28 (5) ◽  
pp. 1059-1072 ◽  
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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