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

TL;DR

This paper characterizes generalized shift operators in $ ext{ell}^p( au)$ spaces as derivations, identifying conditions under which they are (or are not) derivations, Jordan derivations, or generalized derivations, based on the structure of the shift and associated functions.

Contribution

It provides a complete characterization of when generalized shift operators act as derivations or generalized derivations in $ ext{ell}^p( au)$ spaces, extending the understanding of their algebraic properties.

Findings

A $( ext{psi}, ext{lambda})$-derivation occurs iff specific relations involving a function r exist.

$ ext{psi}$-derivations are characterized by $ ext{psi}=rac{1}{2} ext{sigma}_ ext{phi}$.

Generalized derivations occur only when $ ext{phi}$ is the identity map.

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 $\varphi=id_\tau$.