Characterization of Invariant subspaces in the polydisc

Abstract

We give a complete characterization of invariant subspaces for $(M_{z_1},\dots ,M_{z_n})$ on the Hardy space $H^2({\mathbb{D}}^n)$ over the unit polydisc ${\mathbb{D}}^n$ in ${\mathbb{C}}^n$, $n>1$. In particular, this yields a complete set of unitary invariants for invariant subspaces for $(M_{z_1},\dots ,M_{z_n})$ on $H^2({\mathbb{D}}^n)$, $n>1$. As a consequence, we classify a large class of $n$-tuples, $n>1$, of commuting isometries. All of our results hold for vector-valued Hardy spaces over ${\mathbb{D}}^n$, $n>1$. Our invariant subspace theorem solves the well-known open problem on characterizations of invariant subspaces of the Hardy space over the unit polydisc.

23 pages, revised