Weighted Approximation for Algebras and Modules of Continuous Functions: Real and Self-Adjoint Complex Cases

Nachbin, Léopoldo

doi:10.2307/1970617

Cited by 44 publications

(26 citation statements)

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“…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%

“…The fundamental work on weighted space of continuous scalar-valued functions has been done mainly by Nachbin [5], [6] in 1960's. Since then it has been studied extensively for a variety of problems such as weighted approximation, characterization of the dual space, approximation property etc.…”

Section: Introductionmentioning

confidence: 99%

“…Since then it has been studied extensively for a variety of problems such as weighted approximation, characterization of the dual space, approximation property etc. for both scalar-and vector-valued functions (see, for instance, [1], [2], [5], [6], [7], [11]). Recently Singh and Summers [10] have studied the notion of composition operators on CVb(X, E).…”

Section: Introductionmentioning

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

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Operator-Valued Multiplication Operators on Weighted Function Spaces

Khan¹,

Thaheem²

2002

Demonstratio Mathematica

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“…Since then it has been studied extensively for a variety of problems such as weighted approximation, characterization of the dual space, approximation property, description of inductive limit and of tensor-product, etc for both scalar-and vector-valued functions (for instance see [1][2][3][4][5][8][9][10][11][12][13][14]). Recently Singh and Summers [13] have studied the notion of composition operators on CVo(X, C).…”

Section: Introductionmentioning

confidence: 99%

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

Khan

Thaheem

1995

International Journal of Mathematics and Mathematical Sciences

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ABSTRACT. Let X be a completely regular Hausdorff space, E a topological vector space, V a Nachbin family of weights on X, and CVo(X, E) the weighted space of continuous E-valued functions on X. Let 0 X C be a mapping, f E CVo(X, E) and define Me(f) Of (pointwise). In case E is a topological algebra, p X E is a mapping then define M,(f) pf (pointwise). The main purpose of this paper is to give necessary and sufficient conditions for Me and M, to be the multiplication operators on CV0 (X, E) where E is a general topological space (or a suitable topological algebra) which is not necessarily locally convex. These results generalize recent work of Singh and Manhas based on the assumption that E is locally convex.

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“…Introduction. Interest in weighted spaces of continuous functions dates back at least to the Bernstein approximation problem [1], while Nachbin [8], [9] has made more recent use of these spaces in approximation theory. Moreover, weighted spaces provide a general setting for the study of virtually all continuous function spaces encountered in analysis (leading to new results, as well as new, more elegant proofs of known results).…”

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confidence: 99%

A representation theorem for biequicontinuous completed tensor products of weighted spaces

Summers¹

1969

Trans. Amer. Math. Soc.

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1. Introduction. Interest in weighted spaces of continuous functions dates back at least to the Bernstein approximation problem [1], while Nachbin [8], [9] has made more recent use of these spaces in approximation theory. Moreover, weighted spaces provide a general setting for the study of virtually all continuous function spaces encountered in analysis (leading to new results, as well as new, more elegant proofs of known results).In this paper we will be mainly concerned with obtaining a representation of the biequicontinuous completed tensor product of two weighted spaces as another weighted space (via a topological isomorphism). Among its several corollaries, this theorem includes Grothendieck's result [6] for C0(X) ®Ä C0(Y), establishes an analogous result for the space (Cb(X), ß) of bounded continuous functions with the strict topology (see Buck [2]), and leads to the discovery of some interesting new spaces of continuous functions on Xx Y.The technique for obtaining this representation, however, entails an investigation of weighted spaces as locally convex spaces, and this leads to several results of independent interest. In particular, we are able to characterize the topological dual spaces of weighted spaces, and this characterization (which has the flavor of a factorization theorem) has several applications other than those considered here

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Weighted Approximation for Algebras and Modules of Continuous Functions: Real and Self-Adjoint Complex Cases

Cited by 44 publications

References 5 publications

Operator-Valued Multiplication Operators on Weighted Function Spaces

Operator-Valued Multiplication Operators on Weighted Function Spaces

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

A representation theorem for biequicontinuous completed tensor products of weighted spaces

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