Let $H^2$ be the Hardy space and let $A$ be a bounded operator on $H^2$ such that Lat $U\subseteq$ Lat $A$ where $U$ is the unilateral shift. Show that $AU=UA$. (Here Lat $A$ denotes the set of all invariant subspaces of $H^2$ under $A$.)
I am not getting any idea on how to approach this problem. There is a hint to consider adjoints. But I can't link the hint to the problem. Any help is welcome!