Jordan Higher All-Derivable Points in Nest Algebras

TL;DR

This paper proves that in nest algebras, every point is a Jordan higher all-derivable point, meaning all Jordan higher derivable mappings at that point are actual higher derivations, extending previous results to higher derivations.

Contribution

It extends prior work by showing that any point in nest algebras is a Jordan higher all-derivable point, encompassing higher derivations.

Findings

Every point in Alg$ abla$ is a Jordan higher all-derivable point.

Jordan higher derivable mappings at these points are actual higher derivations.

The result generalizes previous findings to higher derivations.

Abstract

Let $\mathcal{N}$ be a non-trivial and complete nest on a Hilbert space $H$. Suppose $d=\{d_n: n\in N\}$ is a group of linear mappings from Alg$\mathcal{N}$ into itself. We say that $d=\{d_n: n\in N\}$ is a Jordan higher derivable mapping at a given point $G$ if $d_{n}(ST+ST)=\sum\limits_{i+j=n}\{d_{i}(S)d_{j}(T)+d_{j}(T)d_{i}(S)\}$ for any $S,T\in Alg \mathcal{N}$ with $ST=G$. An element $G\in Alg \mathcal{N}$ is called a Jordan higher all-derivable point if every Jordan higher derivable mapping at $G$ is a higher derivation. In this paper, we mainly prove that any given point $G$ of Alg$\mathcal{N}$ is a Jordan higher all-derivable point. This extends some results in \cite{Chen11} to the case of higher derivations.