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 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 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

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 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 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 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 Matrix transformations between the sequence space BVP and certain BK spaces

2002 ◽  
Vol 123 (27) ◽  
pp. 33-46 ◽  
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.

scholarly journals Some geometric properties of a sequence space related to ℓp

2003 ◽  
Vol 67 (2) ◽  
pp. 343-347 ◽  
Author(s):  
Mursaleen
Keyword(s):  
Explicit Formula ◽  
Hausdorff Measure ◽  
Sequence Space ◽  
Measure Of Noncompactness ◽  
Bounded Subset ◽  
Hausdorff Measure Of Noncompactness ◽  
Geometric Properties

The sequence space m (ø), introduced and studied by W.L.C. Sargent in 1960, is closely related to the space ℓp. In this paper we obtain an explicit formula for the Hausdorff measure of noncompactness of any bounded subset in m (ø). We also show that m (ø)enjoys the weak Banach-Saks property, while C (m (ø)) = 2. This shows that the condition C (X) < 2, known to be sufficient for the space X to have the weak Banach-Saks property, is not a necessary one.

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