Is there any nontrivial compact generalized shift operator on Hilbert spaces?

Abstract

In the following text for cardinal number \(\tau >0\), and self-map \(\varphi :\tau \rightarrow \tau \) we show the generalized shift operator \(\sigma _\varphi (\ell ^2(\tau ))\subseteq \ell ^2(\tau )\) (where \(\sigma _\varphi ((x_\alpha )_{\alpha<\tau })=(x_{\varphi (\alpha )})_{\alpha <\tau }\) for \((x_\alpha )_{\alpha <\tau }\in {{\mathbb {C}}}^\tau \)) if and only if \(\varphi :\tau \rightarrow \tau \) is bounded and in this case Open image in new window is continuous, consequently Open image in new window is a compact operator if and only if \(\tau \) is finite.