Characterizations of left derivable maps at non-trivial idempotents on nest algebras

TL;DR

This paper characterizes continuous linear maps on nest algebras that are left derivable at non-trivial idempotents, showing they are (generalized) Jordan left derivations and providing specific characterizations under certain conditions.

Contribution

It introduces a characterization of linear maps that are left derivable at non-trivial idempotents on nest algebras, extending understanding of their structure.

Findings

Such maps are (generalized) Jordan left derivations.

Characterizations depend on properties of the map at idempotents.

Results apply to maps continuous in the strong operator topology.

Abstract

Let $Alg \mathcal{N}$ be a nest algebra associated with the nest $ \mathcal{N}$ on a (real or complex) Banach space $\X$. Suppose that there exists a non-trivial idempotent $P\in Alg\mathcal{N}$ with range $P(\X) \in \mathcal{N}$ and $\delta:Alg\mathcal{N} \rightarrow Alg\mathcal{N}$ is a continuous linear mapping (generalized) left derivable at $P$, i.e. $\delta(ab)=a\delta(b)+b\delta(a)$ ($\delta(ab)=a\delta(b)+b\delta(a)-ba\delta(I)$) for any $a,b\in Alg\mathcal{N}$ with $ab=P$. we show that $\delta$ is a (generalized) Jordan left derivation. Moreover, we characterize the strongly operator topology continuous linear maps $\delta$ on some nest algebra $Alg\mathcal{N}$ with property that $\delta(P)=2P\delta(P)$ or $\delta(P)=2P\delta(P)-P\delta(I)$ every idempotent $P$ in $Alg\mathcal{N}$.