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

Abstract

This paper is a follow-up contribution to our work [20] where we discussed some invariant subspace results for contractions on Hilbert spaces. Here we extend the results of [20] to the context of n-tuples of bounded linear operators on Hilbert spaces. Let T = (T_1, \ldots, T_n) be a pure commuting co-spherically contractive n-tuple of operators on a Hilbert space \mathcal{H} and \mathcal{S} be a non-trivial closed subspace of \mathcal{H}. One of our main results states that: \mathcal{S} is a joint T-invariant subspace if and only if there exists a partially isometric operator \Pi \in \mathcal{B}(H^2_n(\mathcal{E}), \mathcal{H})$ such that $\mathcal{S} = \Pi H^2_n(\mathcal{E})$, where H^2_n is the Drury-Arveson space and \mathcal{E} is a coefficient Hilbert space and T_i \Pi = \Pi M_{z_i}, i = 1, \ldots, n. In particular, our work addresses the case of joint shift invariant subspaces of the Hardy space and the weighted Bergman spaces over the unit ball in \mathbb{C}^n.