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

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($\mathcal{E}$), $\mathcal{H}$)$suchthat$$\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.