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 is continuous, consequently is a compact operator if and only if $\tau$ is finite.