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 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 $\mathrm{\Pi }:{\mathrm{H}}_{\mathcal{D}}^{2}\left(\mathrm{\mathbb{D}}\right)\to \mathcal{H}$ such that ΠMz = TΠ and that S = ran Π, or equivalently, PS = ΠΠ*. 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.