  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.
