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Contractive Hilbert modules and their dilations

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

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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 Szego 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 Muller [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 a'', (m) . Some consequences of this more general result are then explored in the case of several natural function algebras.

Item Type: Journal Article
Publication: Israel Journal of Mathematics
Publisher: Hebrew Univ Magnes Press
Additional Information: Copyright of this article belongs to Hebrew Univ Magnes Press.
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Date Deposited: 04 Apr 2012 09:50
Last Modified: 04 Apr 2012 09:50
URI: http://eprints.iisc.ac.in/id/eprint/44185

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