Equivalence classes of mixed invariant subspaces over the bidisk

Abstract

A closed subspace $N$ of the Hardy space $H^2$ over the bidisk is said to be mixed invariant under $T_z$ and $T^\ast_w$ if $T_zN \subset N$ and $T^\ast_w N \subset N$. In this paper, we study unitary, similar and quasi-similar module maps for mixed invariant subspaces. We give some characterization of these maps. All unitary module maps are multiplication operators of unimodular functions. Under the condition $\dim(N \ominus zN)$ = 1, we can describe similar and quasi-similar module maps by outer functions.