Generalized shifts through derivations' concept in $\ell^p(τ)$ 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 $\varphi<sup>{-1}(\alpha)$ has at most $N$ elements for each $\alpha&lt;\tau$, and operators $\psi,\lambda:\ell<sup>p\tau)\to\ell<sup>p(\tau)$ we prove the generalized shift $\mathop{\sigma_\varphi\restriction_{\ell^{p(\tau)}:\ell^{p(\tau)\to\ell^{p(\tau):::::::::}\limits_{:::::::::}}} (x_\alpha)<em>{\alpha&lt;\tau}\mapsto (x</em>{\varphi(\alpha)})<em>{\alpha&lt;\tau}}$: $\bullet$ is a $(\psi,\lambda)-$derivation if and only if there exists $\mathsf{r}\in{\mathbb C}<sup>\tau$ with $$\psi={\mathsf r}\sigma</em>\varphi\restriction_{\ell<sup>p(\tau)}$$ and $\lambda=((1)<em>{\alpha&lt;\tau}-{\mathsf r})\sigma</em>\varphi\restriction_{\ell<sup>p(\tau)}$, $\bullet$ is a $\psi-$derivation if and only if $$\psi=\frac12\sigma_\varphi\restriction_{\ell<sup>p(\tau)}$$, $\bullet$ is not a (Jordan, Jordan triple) derivation, $\bullet$ is a generalized (Jordan, Jordan triple) derivation if and only if $\varphi=id_\tau$.