Extended Composition Operators in Weighted Spaces

L. Oubbi

Abstract: Let $X$ and $Y$ be Hausdorff completely regular spaces and $\beta X$ the Stone-Cech compactification of $X$. For locally convex spaces $E$ and $F$ consisting of continuous functions respectively on $X$ and $Y$ and whose topologies are generated by seminorms that are weighted analogues of the suprimum norm, we give necessary and sufficient conditions for a linear mapping $T: E\to F$ to be an extended composition operator. This means that there exists some map $\varphi: Y\to\beta X$ so that $T(f)=C_{\varphi}(f)$ $(:=\tilde{f}\circ\varphi)$, $f\in E$. Here $\tilde{f}$ stands for the Stone extension of $f$. We also characterize those maps $\varphi$ for which $C_\varphi$ satisfies one of the following conditions:
{\bf 1}) $C_{\varphi}$ is continuous;
{\bf 2}) $C_{\varphi}$ maps some 0-neighbourhood into a bounded set;
{\bf 3}) $C_{\varphi}$ maps some 0-neighbourhood into an equicontinuous, a compact or a weakly compact set;
{\bf 4}) $C_{\varphi}$ maps any bounded set into an equicontinuous, a compact or a weakly compact set.

Keywords: Extended composition operator; (locally) compact operator; strongly bounded operator; (locally) equicontinuous operator; weighted space.

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

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