Rank-one operators in reflexive one-sided A-submodules

Abstract

In this paper, we first characterize reflexive one-sided $\a$-submodules $\u$ of a unital operator algebra $\a$ in $\bh$ completely. Furthermore we investigate the invariant subspace lattice $\lat\r$ and the reflexive hull $\ref\r$, where $\r$ is the submodule generated by rank-one operators in $\u$; in particular, if $\l$ is a subspace lattice, we obtain when the rank-one algebra $\r$ of $\alg\l$ is big enough to determined $\alg\l$ in the following senses: $\alg\l=\alg\lat\r$ and $\alg\l=\ref\r$.