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 $\phi :\tau \to \tau$ if there exists $N\in \mathbb{N}$ such that ${\phi }^{-1}(\alpha )$ has at most $N$ elements for each $\alpha <\tau$, and operators $\psi ,\lambda :{\ell }^p\tau )\to {\ell }^p(\tau )$ we prove the generalized shift $\underset{(x_{\alpha }{)}_{\alpha <\tau }\mapsto (x_{\phi (\alpha )}{)}_{\alpha <\tau }}{{\sigma }_{\phi }{\upharpoonright }_{{\ell }^p(\tau )}:{\ell }^p(\tau )\to {\ell }^p(\tau )}$: $\bullet$ is a $(\psi ,\lambda )-$derivation if and only if there exists $\mathsf{r}\in {\mathbb{C}}^{\tau }$ with $\psi =\mathsf{r}{\sigma }_{\phi }{\upharpoonright }_{{\ell }^p(\tau )}$ and $\lambda =((1{)}_{\alpha <\tau }-\mathsf{r}){\sigma }_{\phi }{\upharpoonright }_{{\ell }^p(\tau )}$, $\bullet$ is a $\psi -$derivation if and only if $\psi =\frac{1}{2}{\sigma }_{\phi }{\upharpoonright }_{{\ell }^p(\tau )}$, $\bullet$ is not a (Jordan, Jordan triple) derivation, $\bullet$ is a generalized (Jordan, Jordan triple) derivation if and only if $\phi =i{d}_{\tau }$.

6 pages