An Invariant Subspace Theorem and Invariant Subspaces of Analytic Reproducing Kernel Hilbert Spaces - I

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Let T be a C· ₀-contraction on a Hilbert space H and S be a non-trivial 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 Π: H²D(D) H such that ΠMz = T Πand that S = ran Π, or equivalently, Pₛ = ΠΠ*. As an application we completely classify the shift-invariant subspaces of C· ₀-contractive and analytic reproducing kernel Hilbert spaces over the unit disc. Our results also includes the case of weighted Bergman spaces over the unit disk.