scholarly journals Two ways to compactness

Filomat ◽  
2003 ◽  
pp. 15-21 ◽  
Author(s):  
Ivana Djolovic
Keyword(s):  
Measure Of Noncompactness ◽  
Linear Operators ◽  
Sequence Spaces ◽  
The Other ◽  
Matrix Transformations

In this paper we give two different ways of proving the compactness of some linear operators between certain sequence spaces. One of them is based only on the theory of matrix transformations and the other uses the Hausdorf 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 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 ON GENERALIZED FIBONACCI DIFFERENCE SPACE DERIVED FROM THE ABSOLUTELY p− SUMMABLE SEQUENCE SPACES

2019 ◽  
pp. 903
Author(s):  
Gülsen Kılınç
Keyword(s):  
Sequence Space ◽  
Bounded Sequence ◽  
Sequence Spaces ◽  
The Other ◽  
Matrix Transformations ◽  
Real Numbers ◽  
Positive Real ◽  
Topological Features ◽  
Summable Sequence ◽  
Alpha Beta

In this study, it is specified \emph{the sequence space} $l\left( F\left( r,s\right),p\right) $, (where $p=\left( p_{k}\right) $ is any bounded sequence of positive real numbers) and researched some algebraic and topological features of this space. Further, $\alpha -,$ $\beta -,$ $\gamma -$ duals and its Schauder Basis are given. The classes of \emph{matrix transformations} from the space $l\left( F\left( r,s\right) ,p\right) $ to the spaces $l_{\infty },c,$ and $% c_{0}$ are qualified. Additionally, acquiring qualifications of some other \emph{matrix transformations} from the space $l\left( F\left( r,s\right) ,p\right) $ to the \emph{Euler, Riesz, difference}, etc., \emph{sequence spaces} is the other result of the paper.

scholarly journals On Some Properties of New Paranormed Sequence Space of Nonabsolute Type

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Vatan Karakaya ◽  
Necip Şimşek
Keyword(s):  
Sequence Space ◽  
Sequence Spaces ◽  
The Other ◽  
Matrix Transformations ◽  
Topological Properties ◽  
Paranormed Sequence Space ◽  
Inclusion Relations

We introduce some new generalized sequence space related to the space . Furthermore we investigate some topological properties as the completeness, the isomorphism, and also we give some inclusion relations between this sequence space and some of the other sequence spaces. In addition, we compute -, -, and -duals of this space and characterize certain matrix transformations on this sequence space.

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 Topological properties and matrix transformations of certain ordered generalized sequence spaces

1995 ◽  
Vol 18 (2) ◽  
pp. 341-356 ◽  
Author(s):  
Manjul Gupta ◽  
Kalika Kaushal
Keyword(s):  
Sequence Space ◽  
Topological Structure ◽  
Necessary Conditions ◽  
Riesz Space ◽  
Linear Operators ◽  
Sequence Spaces ◽  
Matrix Transformations ◽  
Topological Properties ◽  
Vector Valued ◽  
Locally Convex

In this note, we carry out investigations related to the mixed impact of ordering and topological structure of a locally convex solid Riesz space(X,τ)and a scalar valued sequence spaceλ, on the vector valued sequence spaceλ(X)which is formed and topologized with the help ofλandX, and vice versa. Besides,we also characterizeo-matrix transformations fromc(X),ℓ∞(X)to themselves,cs(X)toc(X)and derive necessary conditions for a matrix of linear operators to transformℓ1(X)into a simple ordered vector valued sequence spaceΛ(X).

scholarly journals Matrix transformations involving certain Banach space valued sequence spaces

2000 ◽  
Vol 31 (2) ◽  
pp. 85-100
Author(s):  
J. K. Srivastava ◽  
B. K. Srivastava
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Linear Operators ◽  
Sequence Spaces ◽  
Matrix Transformations ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Matrix Classes ◽  
The Matrix ◽  
Matrix Class

In this paper for Banach spaces $X$ and $Y$ we characterize matrix classes $ (\Gamma (X,\lambda)$, $ l_\infty(Y,\mu))$, $ (\Gamma(X,\lambda),C(Y,\mu))$, $ (\Gamma(X,\lambda)$, $ c_0(Y,\mu))$, $ (\Gamma(X,\lambda)$, $ \Gamma^*(Y,\mu))$, $ (l_1(X,\lambda)$, $ \Gamma(Y,\mu))$ and $ (c_0(X,\lambda)$, $ c_0(Y,\mu))$ of bounded linear operators involving $ X$- and $ Y$-valued sequence spaces. Further as an application of the matrix class $ (c_0(X,\lambda)$, $ c_0(Y,\mu))$ we investigate the Banach space $ B(c_0(X,\lambda)$, $ c_0(Y,\mu))$ of all bounded linear mappings of $ c_0(x,\lambda)$ to $ c_0(Y,\mu)$.

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

Filomat ◽  
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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