AN INVARIANT SUBSPACE THEOREM AND INVARIANT SUBSPACES OF ANALYTIC REPRODUCING KERNEL HILBERT SPACES. I

Let T be a C.(0)-contraction on a Hilbert space H and S be a nontrivial closed subspace of H. We prove that S is a T-invariant subspace of H if and only if there exists a Hilbert space D and a partially isometric operator Pi: H-D(2)(D) -> H such that Pi M-z = T Pi and that S = ran Pi, or equivalently, P-S = Pi Pi*. As an application we completely classify the shift-invariant subspaces of analytic reproducing kernel Hilbert spaces over the unit disc. Our results also include the case of weighted Bergman spaces over the unit disk.