Operator-Valued Multiplication Operators on Weighted Function Spaces

Abstract

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,

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

“…The purpose of this paper is to characterize those weighted composition operators on non-locally convex weighted spaces which are induced by mappings π : X → B(E) and T : X → X. These results improve and extend, in particular, some of the results contained in [4,6,9,10,12,14].…”

“…In the locally convex setting, a characterization of WCO on CV b (X, E) has been presented by Singh and one of the authors in [10] under the assumption that π(X) is equicontinuous whereas on CV 0 (X, E) it has been reported by Manhas in [6] but under the condition that X is a k R -space. For non-locally convex spaces, multiplication operators on weighted spaces have been studied by Khan and Thaheem [4] with the same requirement as in [6]. It can be noted that either of the condition that "π(X) is equicontinuous" or "X is a k R -space" is needed only to make the map M π continuous (cf.…”

“…[15]). Also, the results in [4,6,10,14] characterizing WCOs and multiplication operators on weighted spaces require as a hypothesis that T is continuous on whole of X and π is continuous in the uniform operator topology. But for instance when π(x) = 0 for all x ∈ X, W π,T is a continuous operator even if T is not continuous on whole of X.…”

“…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. Khan and Thaheem, in a very recent work [4], have extended the work of [14] to a non-locally convex setting and their work have been further extended by Singh and Kour [8]. This paper is a continuation of earlier paper [12] in which a characterization of weighted composition operators on locally convex weighted space CV b (X, E) is presented and also it is a continuation to earlier paper [9] where we have studied weighted composition operators W π,T (W θ,T ) on non-locally convex weighted spaces CV b (X, E) and CV 0 (X, E) induced by π : X → E(θ : X → C) and T : X → X.…”

“…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.…”