Operator positivity and analytic models of commuting tuples of operators

Article Type

Research Article

Publication Title

Studia Mathematica

Abstract

We study analytic models of operators of class C.0 with natural positivity assumptions. In particular, we prove that for an m-hypercontraction T ∈ C.0 on a Hilbert space ℋ, there exist Hilbert spaces ϵ and ϵ∗ and a partially isometric multiplier θ ∈ ℳ(H2(ϵ),A2m(ϵ∗)) such that ℋ ≅ Qθ = A2m(ϵ∗) ⊖ θH2(ϵ) and T ≅ PQθMz|Qθ, where A2m(ϵ∗) is the ϵ∗-valued weighted Bergman space and H2(ϵ) is the ϵ-valued Hardy space over the unit disc D. We then proceed to study analytic models for doubly commuting n-tuples of operators and investigate their applications to joint shift co-invariant subspaces of reproducing kernel Hilbert spaces over the polydisc. In particular, we completely analyze doubly commuting quotient modules of a large class of reproducing kernel Hilbert modules, in the sense of Arazy and Engliš, over the unit polydisc Dn.

First Page

155

Last Page

171

DOI

10.4064/sm8437-2-2016

Publication Date

1-1-2016

Comments

Open Access; Green Open Access

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