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 $d_{n}(ST+TS) = \sum\limits_{i+j=n} \{d_{i}(S) d_{j}(T) + d_{j}(T) d_{i}(S)\}$ 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 [1] to the case of higher derivations.