On a class of left ideals of nest algebras

Abstract

We introduce a class of left ideals (and subalgebras) of nest algebras determined by totally ordered families of partial isometries on a complex Hilbert space $H$. Let $\mathcal{E}$ be a family of partial isometries that is totally ordered in the Halmos--McLaughlin ordering, and let $\mathcal{A}_{\mathcal{E}}$ be the subset of operators in $B(H)$ which, for all $E\in \mathcal{E}$, map the initial space of $E$ to the final space of $E$. We show that $\mathcal{A}_{\mathcal{E}}$ is a subalgebra of $B(H)$ if and only if $\mathcal{A}_{\mathcal{E}}$ is a left ideal of a certain nest algebra, and if so, $\mathcal{E}$ consists of power partial isometries, except possibly for its supremum $\vee \mathcal{E}$, in which case the range $\operatorname{ran}(\vee \mathcal{E})$ is $H$. It is also shown that any left ideal $\mathcal{A}_{\mathcal{E}}$ is decomposable and that the subset of finite rank operators in its closed unit ball is strongly dense in the ball. Necessary and sufficient conditions to solve $Tx=y$ and $T^*x=y$ in $\mathcal{A}_{\mathcal{E}}$ are given.