Contractive Hilbert modules and their dilations |
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Authors: | Ronald G Douglas Gadadhar Misra Jaydeb Sarkar |
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Institution: | 1.Department of Mathematics,Texas A&M University College Station,Texas,USA;2.Department of Mathematics,Indian Institute of Science,Bangalore,India |
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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 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 ? m . Some consequences of this more general result are then explored in the case of several natural function algebras. |
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