Contractive Hilbert modules and their dilations

TL;DR

This paper characterizes when a quasi-free Hilbert module over the polydisk admits a unique minimal dilation to the Hardy module, linking kernel positivity to a specific factorization involving the Szeg"o kernel.

Contribution

It establishes a new criterion for dilations of Hilbert modules based on kernel factorizations and extends the theory to more general quasi-free modules.

Findings

Dilation exists iff S^{-1}k is positive

Explicit dilation realization provided

Results extend to broader classes of modules

Abstract

In this note, we show that a quasi-free Hilbert module R defined over the polydisk algebra with kernel function k(z, w) admits a unique minimal dilation (actually an isometric co-extension) to the Hardy module over the polydisk if and only if S^{-1}(z, w) k(z, w) is a positive kernel function, where S(z, w) is the Szeg\"{o} kernel for the polydisk. Moreover, we establish the equivalence of such a factorization of the kernel function and a positivity condition, defined using the hereditary functional calculus, which was introduced earlier by Athavale \cite{Ath} and Ambrozie, Englis and M\"{u}ller. An explicit realization of the dilation space is given along with the isometric embedding of the module R in it. The proof works for a wider class of Hilbert modules in which the Hardy module is replaced by more general quasi-free Hilbert modules such as the classical spaces on the polydisk or the unit ball in {C}^m. Some consequences of this more general result are then explored in the case of several natural function algebras.