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

Abstract

In the following text for cardinal number $\tau>0$, and self--map $\varphi:\tau\to\tau$ 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 $\varphi:\tau\to\tau$ is bounded and in this case $\sigma_\varphi\restriction_{\ell^2(\tau)}:\ell^2(\tau)\to\ell^2(\tau)$ is continuous, consequently $\sigma_\varphi\restriction_{\ell^2(\tau)}:\ell^2(\tau)\to\ell^2(\tau)$ is a compact operator if and only if $\tau$ is finite.