Generalized shifts through derivations' concept in $\ell^p(τ)$ spaces
Safoura Arzanesh, Fatemah Ayatollah Zadeh Shirazi, Arezoo Hosseini
Abstract
In the following text for $p\in[1,\infty]$, nonzero cardinal number $τ$, self--map $\varphi:τ\toτ$ if there exists $N\in\mathbb{N}$ such that $\varphi^{-1}(α)$ has at most $N$ elements for each $α<τ$, and operators $ψ,λ:\ell^pτ)\to\ell^p(τ)$ we prove the generalized shift $\mathop{σ_\varphi\restriction_{\ell^p(τ)}:\ell^p(τ)\to\ell^p(τ)\:\:\:\:\:\:\:\:\:}\limits_{\:\:\:\:\:\:\:\:\: (x_α)_{α<τ}\mapsto (x_{\varphi(α)})_{α<τ}}$: $\bullet$ is a $(ψ,λ)-$derivation if and only if there exists $\mathsf{r}\in{\mathbb C}^τ$ with $ψ={\mathsf r}σ_\varphi\restriction_{\ell^p(τ)}$ and $λ=((1)_{α<τ}-{\mathsf r})σ_\varphi\restriction_{\ell^p(τ)}$, $\bullet$ is a $ψ-$derivation if and only if $ψ=\frac12σ_\varphi\restriction_{\ell^p(τ)}$, $\bullet$ is not a (Jordan, Jordan triple) derivation, $\bullet$ is a generalized (Jordan, Jordan triple) derivation if and only if $\varphi=id_τ$.