Multiplication Operators and Dynamical Systems

Abstract

Let X be a completely regular Hausdorff space, let V be a system of weights on X and let T be a locally convex Hausdorff topological vector space. Then CV b (X, T) is a locally convex space of vector-valued continuous functions with a topology generated by seminorms which are weighted analogues of the supremum norm. In the present paper we characterize multiplication operators on the space CV b (X, T) induced by operator-valued mappings and then obtain a (linear) dynamical system on this weighted function space.1991 Mathematics subject classification (Amer. Math. Soc.): 34 C 35, 46 E 40, 47 B 38.

Operator-Valued Multiplication Operators on Weighted Function Spaces

“…These results extend, in particular, some results of Singh and Manhas [9] for E a locally convex space or a normed space to the case of E a general space or a locally bounded space, respectively.…”

“…The following result with E being a locally bounded space extends ( [9], Proposition 2.3) where E has been taken to be a normed space. THEOREM …”

“…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].…”

“…Recently Singh and Summers [10] have studied the notion of composition operators on CVb(X, E). 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.…”

Corrigendum to Multiplication operators and dynamical systems

“…Throughout Section 2 of [1], X should be taken as a completely regular Hausdorff A" R -space. The lines 1 through 26 in the proof of Theorem 2.1 of the paper should be replaced by the following lines.…”