Characterization of Invariant subspaces in the polydisc

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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), 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 Dⁿ, 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.