Characterizing derivations for any nest algebras on Banach spaces by their behaviors at an injective operator

TL;DR

This paper characterizes all-derivable points in nest algebras on Banach spaces, showing that injective operators and operators with dense range are such points without extra nest assumptions.

Contribution

It establishes that injective and dense-range operators are all-derivable points in nest algebras on Banach spaces, generalizing previous results.

Findings

Injective operators are all-derivable points.

Operators with dense range are all-derivable points.

No additional nest assumptions are needed.

Abstract

Let ${\mathcal N}$ be a nest on a complex Banach space $X$ and let $\mbox{ Alg}{\mathcal N}$ be the associated nest algebra. We say that an operator $Z\in \mbox{ Alg}{\mathcal N}$ is an all-derivable point of $\mbox{ Alg}{\mathcal N}$ if every linear map $\delta$ from $\mbox{ Alg}{\mathcal N}$ into itself derivable at $Z$ (i.e. $\delta$ satisfies $\delta(A)B+A\delta(B)=\delta(Z)$ for any $A,B \in \mbox{ Alg}{\mathcal N}$ with $AB=Z$) is a derivation. In this paper, it is shown that every injective operator and every operator with dense range in $\mbox{Alg}{\mathcal N}$ are all-derivable points of $\mbox{Alg}{\mathcal N}$ without any additional assumption on the nest.