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

TL;DR

This paper characterizes when generalized shift operators on Hilbert spaces are bounded, continuous, or compact, concluding that nontrivial compact operators only occur on finite-dimensional spaces.

Contribution

It provides a complete characterization of boundedness, continuity, and compactness of generalized shift operators on Hilbert spaces based on the properties of the underlying self-map.

Findings

Generalized shift operators are bounded iff the underlying map is bounded.

Such operators are compact only on finite-dimensional spaces.

The paper establishes the equivalence between boundedness, continuity, and the structure of the underlying map.

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.