Generalized shifts through derivations' concept in $\ell^p(\tau)$ spaces

Abstract

In the following text for $p\in[1,\infty]$, nonzero cardinal number $\tau$, self--map $\varphi:\tau\to\tau$ if there exists $N\in\mathbb{N}$ such that $\varphi^{-1}(\alpha)$ has at most $N$ elements for each $\alpha<\tau$, and operators $\psi,\lambda:\ell^p\tau)\to\ell^p(\tau)$ we prove the generalized shift $\mathop{\sigma_\varphi\restriction_{\ell^p(\tau)}:\ell^p(\tau)\to\ell^p(\tau)\:\:\:\:\:\:\:\:\:}\limits_{\:\:\:\:\:\:\:\:\: (x_\alpha)_{\alpha<\tau}\mapsto (x_{\varphi(\alpha)})_{\alpha<\tau}}$: $\bullet$ is a $(\psi,\lambda)-$derivation if and only if there exists $\mathsf{r}\in{\mathbb C}^\tau$ with $\psi={\mathsf r}\sigma_\varphi\restriction_{\ell^p(\tau)}$ and $\lambda=((1)_{\alpha<\tau}-{\mathsf r})\sigma_\varphi\restriction_{\ell^p(\tau)}$, $\bullet$ is a $\psi-$derivation if and only if $\psi=\frac12\sigma_\varphi\restriction_{\ell^p(\tau)}$, $\bullet$ is not a (Jordan, Jordan triple) derivation, $\bullet$ is a generalized (Jordan, Jordan triple) derivation if and only if $\varphi=id_\tau$.