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

Abstract

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

5 pages