Characterization of invariant subspaces in the polydisc

Abstract

We give a complete characterization of invariant subspaces for (Mz1,…,Mzn) on the Hardy space H2(Dn) over the unit polydisc Dn in Cn, n>1. In particular, this yields a complete set of unitary invariants for invariant subspaces for (Mz1,…,Mzn) on H2(Dn). As a consequence, we classify a large class of n-tuples of commuting isometries. All of our results hold for vector-valued Hardy spaces over Dn, n>1.