Characterizations of all-derivable points in nest algebras

TL;DR

This paper characterizes all-derivable points in nest algebras, showing that any non-zero element in the algebra is an all-derivable point, thus providing a complete description of such points.

Contribution

It establishes a precise criterion for all-derivable points in nest algebras, identifying non-zero elements as exactly those points.

Findings

All non-zero elements in nest algebras are all-derivable points.

Zero element in nest algebra is not an all-derivable point.

Provides a complete characterization of all-derivable points in nest algebras.

Abstract

Let $\mathcal{A}$ be an operator algebra on a Hilbert space. We say that an element $G\in {\mathcal{A}}$ is an all-derivable point of ${\mathcal{A}}$ if every derivable linear mapping $\phi$ at $G$ (i.e. $\phi(ST)=\phi(S)T+S\phi(T)$ for any $S,T\in alg{\mathcal{N}}$ with $ST=G$) is a derivation. Suppose that $\mathcal{N}$ is a nontrivial complete nest on a Hilbert space $H$. We show in this paper that $G\in {alg\mathcal{N}}$ is an all-derivable point if and only if $G\neq0$.