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 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 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.

Some classes of operators related to the space of convergent series cs

Filomat ◽  
2017 ◽  
Vol 31 (20) ◽  
pp. 6329-6335
Author(s):  
Katarina Petkovic
Keyword(s):  
Hausdorff Measure ◽  
Sequence Space ◽  
Measure Of Noncompactness ◽  
Convergent Series ◽  
Linear Operators ◽  
Hausdorff Measure Of Noncompactness ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Matrix Domain ◽  
Matrix Domains

Sequence space of convergent series can also be seen as a matrix domain of triangle. By using the theory of matrix domains of triangle, as well as the fact that cs is an AK space we can give the representation of some general bounded linear operators related to the cs sequence space. We will also give the conditions for compactness by using the Hausdorff measure of noncompactness.

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.

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.

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