All-derivable points in nest algebras

TL;DR

This paper characterizes all-derivable points in nest algebras on Hilbert spaces, showing that certain invertible elements restricted to subspaces are all-derivable points under the strong operator topology.

Contribution

It provides a new criterion for identifying all-derivable points in nest algebras based on invertibility conditions on subspace restrictions.

Findings

Identifies conditions under which elements are all-derivable points in nest algebras.

Establishes the importance of invertibility on subspaces for derivable mappings.

Extends understanding of derivations in operator algebras.

Abstract

Suppose that $\mathscr{A}$ is an operator algebra on a Hilbert space $H$. An element $V$ in $\mathscr{A}$ is called an all-derivable point of $\mathscr{A}$ for the strong operator topology if every strong operator topology continuous derivable mapping $\phi$ at $V$ is a derivation. Let $\mathscr{N}$ be a complete nest on a complex and separable Hilbert space $H$. Suppose that $M$ belongs to $\mathscr{N}$ with $\{0\}\neq M\neq\ H$ and write $\hat{M}$ for $M$ or $M^{\bot}$. Our main result is: for any $\Omega\in alg\mathscr{N}$ with $\Omega=P(\hat{M})\Omega P(\hat{M})$, if $\Omega |_{\hat{M}}$ is invertible in $alg\mathscr{N}_{\hat{M}}$, then $\Omega$ is an all-derivable point in $alg\mathscr{N}$ for the strong operator topology.