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 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 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 New approach to some results related to mixed norm sequence spaces

Filomat ◽  
2016 ◽  
Vol 30 (1) ◽  
pp. 83-88 ◽  
Author(s):  
Ivana Djolovic ◽  
Eberhard Malkowsky ◽  
Katarina Petkovic
Keyword(s):  
Hausdorff Measure ◽  
Measure Of Noncompactness ◽  
Diagonal Matrix ◽  
Sequence Spaces ◽  
Mixed Norm ◽  
Hausdorff Measure Of Noncompactness ◽  
Infinite Matrices ◽  
New Approach ◽  
The Subject ◽  
Bk Spaces

In this paper, the mixed norm sequence spaces ?p,q for 1 ? p,q ? ? are the subject of our research; we establish conditions for an operator T? to be compact, where T? is given by a diagonal matrix. This will be achieved by applying the Hausdorff measure of noncompactness and the theory of BK spaces. This problem was treated and solved in [5, 6], but in a different way, without the application of the theory of infinite matrices and BK spaces. Here, we will present a new approach to the problem. Some of our results are known and others are new.

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 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 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.

Banach spaces of absolutely k-summable series

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Mehmet Ali Sarıgöl ◽  
Ravi P. Agarwal
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Hausdorff Measure ◽  
Measure Of Noncompactness ◽  
Compact Operators ◽  
Matrix Operator ◽  
Hausdorff Measure Of Noncompactness ◽  
Bk Space ◽  
New Space ◽  
General Banach Space

Abstract In this paper, we present a general Banach space of absolutely k-summable series using a triangle matrix operator and prove that this is a BK-space isometrically isomorphic to the space ℓ k {\ell_{k}} . We also establish the α - {\alpha-} , β - {\beta-} , γ-duals and base of the new space. Finally, we qualify some matrix and compact operators on the new space making use of the Hausdorff measure of noncompactness. Our results include, as particular cases, a number of well-known results.

scholarly journals Compact operators on some generalized mixed norm spaces

Filomat ◽  
2014 ◽  
Vol 28 (5) ◽  
pp. 1019-1026
Author(s):  
A. Alotaibi ◽  
E. Malkowsky ◽  
H. Nergiz
Keyword(s):  
Hausdorff Measure ◽  
Measure Of Noncompactness ◽  
Sufficient Conditions ◽  
Compact Operators ◽  
Necessary And Sufficient Conditions ◽  
Mixed Norm ◽  
Hausdorff Measure Of Noncompactness ◽  
Mixed Norm Spaces ◽  
Necessary And Sufficient

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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