Invariant subspaces in the polydisk

Abstract

This note is a study of unitary equivalence of invariant subspaces of H2 of the polydisk. By definition, this means joint unitary equivalence of the shift operators restricted to the invariant subspaces. In one variable, all invariant subspaces are unitarily equivalent and all can be represented as inner functions times H2. In several variables, our results suggest that unitary equivalence and multiplication by inner functions are again related. For example, all invariant subspaces of a given invariant subspace Jt which are unitarily equivalent to Jί are φJΐ, for φ inner; and all invariant subspaces unitarily equivalent to an invariant subspace Ji of finite codimension are yJί. In particular, two invariant subspaces of finite codimension are unitarily equivalent if and only if they are equal. 0. The classic paper of Beurling [4] led to a spate of research in operator theory, Hp theory, and other areas, which continues to the present. Although Beurling's answer to the problem of spectral synthesis was negative, his explicit characterization of the invariant subspaces of the unilateral shift, in terms of the inner-outer factorization of analytic functions has had a major impact. Since the Hardy space for the unit disk is the primary nontrivial example for so many different areas, it is not surprizing that this characterization has proved so important. Almost everyone who has thought about this topic must have considered the corresponding problem for H2 of the polydisk. Although the existence of inner functions in this context is obvious (in contrast with the case of the ball), one quickly sees that a Beurling-like characterization is not possible. Most results on this problem have gone unpublished [10], except for the book of Rudin [11], and many results are counterexamples. One exception is a characterization, by Ahern and the second author, of invariant subspaces having finite codimension, as the closure of the ideals in C[zv... ,zN] of finite codimension; [2]. In this note, we eschew the goal of characterizin g all invariant subspaces in the polydisk and instead investigate equivalence of invariant subspaces in N variables, under joint unitary equivalence of the restrictions of the N coordinate shifts. Put another way, we shall consider the problem of characterizin g the (inequivalent) submodules of the Hardy space over the polydisk algebra. The existence of nonunitarily equivalent submodules is directly related to the