Jordan Higher All-Derivable Points in Nest Algebras

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.