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

TL;DR

This paper characterizes invariant subspaces of certain contraction operators on Hilbert spaces and classifies shift-invariant subspaces in analytic reproducing kernel Hilbert spaces, including weighted Bergman spaces.

Contribution

It provides a complete characterization of invariant subspaces for C₀-contractions and classifies shift-invariant subspaces in analytic RKHS, extending to weighted Bergman spaces.

Findings

Invariant subspaces correspond to partially isometric operators from vector-valued Hardy spaces.

Complete classification of shift-invariant subspaces in C₀-contractive analytic RKHS.

Results include the case of weighted Bergman spaces over the unit disk.

Abstract

Let T be a C_{\cdot 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 \Pi : H^2_D(\mathbb{D}) \raro 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 C_{\cdot 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.