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

In the following text for $p\in [1,\mathrm{\infty }]$, nonzero cardinal number $\tau$, self–map $\phi :\tau \to \tau$ if there exists $N\in \mathbb{N}$ such that ${\phi }^{-1}(\alpha )$ has at most $N$ elements for each $\alpha <\tau$, and operators $\psi ,\lambda :{\mathrm{\ell }}^p\tau )\to {\mathrm{\ell }}^p(\tau )$ we prove the generalized shift $\mathop{\sigma_\varphi\restriction_{\ell^p(\tau)}:\ell^p(\tau)\to\ell^p(\tau):::::::::}\limits_{::::::::: (x_\alpha){\alpha<\tau}\mapsto (x{\varphi(\alpha)}){\alpha<\tau}}$:$\bullet$isa$(\psi,\lambda)-$derivationifandonlyifthereexists$$\mathsf{r}$\in{\mathbb C}^\tau$with$\psi={\mathsf r}\sigma\varphi\restriction_{\ell^p(\tau)}$and$\lambda=((1){\alpha<\tau}-{\mathsf r})\sigma\varphi\restriction_{\ell^p(\tau)}$,$\bullet$isa$\psi-$derivationifandonlyif$\psi=\frac12\sigma_\varphi\restriction_{\ell^p(\tau)}$,$\bullet$isnota(Jordan,Jordantriple)derivation,$\bullet$isageneralized(Jordan,Jordantriple)derivationifandonlyif$\varphi=id_\tau$.