Contractive Hilbert modules and their dilations

Douglas, Ronald G. ; Misra, Gadadhar ; Sarkar, Jaydeb (2012) Contractive Hilbert modules and their dilations Israel Journal of Mathematics, 187 (1). pp. 141-165. ISSN 0021-2172

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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 Szegö 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 [8] and Ambrozie, Englis and Müller [2]. 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 Cm. Some consequences of this more general result are then explored in the case of several natural function algebras.

Item Type:Article
Source:Copyright of this article belongs to Springer.
ID Code:88254
Deposited On:27 Mar 2012 13:01
Last Modified:27 Mar 2012 13:01

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