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 (Sarkar in J Oper Theory, 73:433–441, 2015) where we discussed some invariant subspace results for contractions on Hilbert spaces. Here we extend the results of (Sarkar in J Oper Theory, 73:433–441, 2015) to the context of n-tuples of bounded linear operators on Hilbert spaces. Let T = (T₁,…,T_n) be a pure commuting co-spherically contractive n-tuple of operators on a Hilbert space H and S be a non-trivial closed subspace of H. One of our main results states that: S is a joint T-invariant subspace if and only if there exists a partially isometric operator Π ∊B(H²_n(ε),H) such that , S = ΠH²_n(ε) where H²_n is the Drury–Arveson space and ε is a coefficient Hilbert space and TiΠ = ΠM_zi, i = 1,…,n. In particular, it follows that a shift invariant subspace of a “nice” reproducing kernel Hilbert space over the unit ball in C_n is the range of a “multiplier” with closed range. Our work addresses the case of joint shift invariant subspaces of the Hardy space and the weighted Bergman spaces over the unit ball in C_n.