Characterization of Invariant subspaces in the polydisc

We give a complete characterization of invariant subspaces for $(M_{z_1}, \ldots, M_{z_n})$ontheHardyspace$H^2($\mathbb{D}$^n)$overtheunitpolydisc$$\mathbb{D}$^n$in$$\mathbb{C}$^n$,$n >1$. In particular, this yields a complete set of unitary invariants for invariant subspaces for $(M_{z_1}, \ldots, M_{z_n})$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.