An invariant subspace theorem and invariant subspaces of analytic reproducing kernel Hilbert spaces. II (Q285494)

This paper is a continuation of the author's work published in [J. Oper. Theory 73, No. 2, 433--441 (2015; Zbl 1389.47029)], where some invariant subspace results for contractions in Hilbert spaces were obtained. In the present paper such results are extended to so-called pure commuting co-spherically contractive \(n\)-tuples of operators on a Hilbert space. Specifically, an invariant subspace \(\mathcal S\) under such an \(n\)-tuple \(T=(T_1,\ldots,T_n)\) is characterized in terms of the existence of a partially isometric operator \(\Pi\in{\mathcal B}(H^2_n({\mathcal E}),{\mathcal H})\) such that \({\mathcal S}=\Pi H^2_n({\mathcal E})\) and \( T_i\Pi=\Pi M_{z_i},\,i=1,S\dots,n\), where \( {\mathcal E}\) is a coefficient space, and \( H^2_n({\mathcal E})\) is a Drury-Arveson space. This result applies to the case of joint shift invariant subspaces of Hardy or weighted Bergman spaces over the unit ball in \({\mathbb C}^n\).