Multiplication Operators on Weighted Spaces of Continuous Functions

L. Oubbi

Abstract: Let $V$ be a Nachbin family on the Hausdorff completely regular space $X$, $E$ a locally convex space, ${\cal B}(E)$ the algebra of all continuous operators on $E$ and $\psi:X\to{\cal B}(E)$ a map. We give necessary and sufficient conditions for the induced linear mapping $M_\psi: f\mapsto\psi(\ )(f(\ ))$ to be a multiplication operator on a subspace of the weighted space of $E$-valued continuous functions $CV(X, E)$. Next, we characterize the bounded multiplication operators and show that, at least whenever $X$ is a $V_{\R}$-space, such an operator is precompact if and only if it is trivial.

Keywords: multiplication operator; Nachbin family; weighted space.

Classification (MSC2000): 47B38, 46E10, 46E25.

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