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

Abstract

In the following text for cardinal number $τ>0$, and self--map $φ:τ\toτ$ we show the generalized shift operator $σ_φ(\ell^2(τ))\subseteq\ell^2(τ)$ (where $σ_φ((x_α)_{α<τ})=(x_{φ(α)})_{α<τ}$ for $(x_α)_{α<τ}\in{\mathbb C}^τ$) if and only if $φ:τ\toτ$ is bounded and in this case $σ_φ\restriction_{\ell^2(τ)}:\ell^2(τ)\to\ell^2(τ)$ is continuous, consequently $σ_φ\restriction_{\ell^2(τ)}:\ell^2(τ)\to\ell^2(τ)$ is a compact operator if and only if $τ$ is finite.