Is there any nontrivial compact generalized shift operator on Hilbert spaces?
Fatemah Ayatollah Zadeh Shirazi, Fatemeh Ebrahimifar
Abstract
In the following text for cardinal number $τ>0$, and self--map $\varphi:τ\toτ$ we show the generalized shift operator $σ_\varphi(\ell^2(τ))\subseteq\ell^2(τ)$ (where $σ_\varphi((x_α)_{α<τ})=(x_{\varphi(α)})_{α<τ}$ for $(x_α)_{α<τ}\in{\mathbb C}^τ$) if and only if $\varphi:τ\toτ$ is bounded and in this case $σ_\varphi\restriction_{\ell^2(τ)}:\ell^2(τ)\to\ell^2(τ)$ is continuous, consequently $σ_\varphi\restriction_{\ell^2(τ)}:\ell^2(τ)\to\ell^2(τ)$ is a compact operator if and only if $τ$ is finite.