Multiplication operators on weighted spaces in the non‐locally convex framework

Khan, Liaqat Ali; Thaheem, A. B.

doi:10.1155/s0161171297000112

Cited by 3 publications

(6 citation statements)

References 14 publications

(15 reference statements)

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“…This paper is a continuation of our earlier paper [3] in which we have studied, in the non-locally convex framework, multiplication operators on weighted function spaces which are induced by scalar-and vector-valued mappings.…”

Section: Introductionmentioning

confidence: 92%

“…Later, Singh and Manhas [8], [9] made an analogous study of multiplication operators on CV¡, (X, E), assuming E to be a locally convex space or a normed space. This paper is a continuation of our earlier paper [3] in which we have studied, in the non-locally convex framework, multiplication operators on weighted function spaces which are induced by scalar-and vector-valued mappings.…”

Section: Introductionmentioning

confidence: 92%

“…We mention that if V = S¿(X), then CV b (X,E) = CV 0 (X,E) = Cb(X, E) and w v = β, the strict topology, and we write it as (Cb(X, Ε),β)\ if V = S+(X), then CV b {X,E) = CV 0 (X,E) = C{X,E) and w v = k, the compact-open topology, and we write it as (C(X, E),k). For more information on weighted spaces, we refer to [1][2][3], [5][6][7][8][9][10][11].…”

Section: N(v G) = {F β CV B (X E) : (Vf)(x) C G}mentioning

confidence: 99%

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Operator-Valued Multiplication Operators on Weighted Function Spaces

Khan¹,

Thaheem²

2002

Demonstratio Mathematica

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Abstract. Let X be a completely regular Hausdorff space, E a Hausdorff topological vector space, V a Nachbin family of weights on X, and CVi, (X,E) the weighted space of continuous /^-valued functions on X. Let B(E) be the vector space of all continuous linear mappings from E into itself, endowed with the topology of uniform convergence on bounded sets. If φ : X -* B(E) is a continuous mapping and / € CV^(X, E), let(χ ζ X). In this paper we give a necessary and sufficient condition for Μφ to be the multiplication operator (i.e. continuous selfmapping) on CV¡, (X, E), where E is a general space or a locally bounded space. These results extend recent work of Singh and Manhas to a non-locally convex setting and that of the authors where φ has been considered to be a complex or ^-valued map. IntroductionThe fundamental work on weighted space of continuous scalar-valued functions has been done mainly by Nachbin (X, E), assuming E to be a locally convex space or a normed space. This paper is a continuation of our earlier paper [3] in which we have studied, in the non-locally convex framework, multiplication operators on weighted function spaces which are induced by scalar-and vector-valued mappings.The purpose of this paper is to characterize those multiplication operators which are induced by operator-valued mappings. These results extend,

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Section: Introductionmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 92%

Section: N(v G) = {F β CV B (X E) : (Vf)(x) C G}mentioning

confidence: 99%

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Operator-Valued Multiplication Operators on Weighted Function Spaces

Khan¹,

Thaheem²

2002

Demonstratio Mathematica

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“…and w V in this case is the substrict topology. For more details on such weighted spaces, we refer to Nachbin [7], Singh and Summers [17], Khan [2] and Khan and Thaheen [3].…”

Section: Preliminariesmentioning

confidence: 99%

“…The contents of this paper are in relation with the theory of weighted composition operators on weighted spaces which are studied by Jamison and Rajagopalan [1], Singh and Summers [17], Singh and Manhas [14], Singh and one of the authors in [10,12], Khan and Thaheem [3,4], Manhas [6], and two of the authors in [8,9]. In [17], Singh and Summers have made a detailed study of composition operators on locally convex weighted spaces where as multiplication operators on such spaces have been studied by Singh and Manhas [14] and their results have been generalized by Singh and Singh [10] to a larger class of operators, known as weighted composition operators.…”

Section: Introductionmentioning

confidence: 99%

On characterizations of weighted composition opeartors on non-locally convex weighted spaces of continuous functions

Sharma¹,

Kour²,

Singh³

2000

Tamkang J. Math.

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For a system $V$ of weights on a completely regular Hausdorff space $X$ and a Hausdorff topological vector space $E$, let $ CV_b(X,E)$ and $ CV_0(X,E)$ respectively denote the weighted spaces of continuouse $E$-valued functions $f$ on $X$ for which $ (vf)(X)$ is bounded in $E$ and $vf$ vanishes at infinity on $X$ all $ v\in V$. On $CV_b(X,E)(CV_0(X,E))$, consider the weighted topology, which is Hausdorff, linear and has a base of neighbourhoods of 0 consising of all sets of the form: $ N(v,G)=\{f:(vf)(X)\subseteq G\}$, where $v\in V$ and $G$ is a neighbourhood of 0 in $E$. In this paper, we characterize weighted composition operators on weighted spaces for the case when $V$ consists of those weights which are bounded and vanishing at infinity on $X$. These results, in turn, improve and extend some of the recent works of Singh and Singh [10, 12] and Manhas [6] to a non-locally convex setting as well as that of Singh and Manhas [14] and Khan and Thaheem [4] to a larger class of operators.

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