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

Article Type

Research Article

Publication Title

Complex Analysis and Operator Theory

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 (Formula presented.) be a pure commuting co-spherically contractive n-tuple of operators on a Hilbert space (Formula presented.) and (Formula presented.) be a non-trivial closed subspace of (Formula presented.). One of our main results states that: (Formula presented.) is a joint T-invariant subspace if and only if there exists a partially isometric operator (Formula presented.) such that (Formula presented.) , where (Formula presented.) is the Drury–Arveson space and (Formula presented.) is a coefficient Hilbert space and (Formula presented.) , (Formula presented.). In particular, it follows that a shift invariant subspace of a “nice” reproducing kernel Hilbert space over the unit ball in (Formula presented.) 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 (Formula presented.).

First Page

769

Last Page

782

DOI

10.1007/s11785-015-0501-8

Publication Date

4-1-2016

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