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Complete Pick positivity and unitary invariance

Bhattacharya, Angshuman and Bhattacharyya, Tirthankar (2010) Complete Pick positivity and unitary invariance. In: Studia Mathematica, 200 (2). pp. 149-162.

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Official URL: http://journals.impan.pl/cgi-bin/doi?sm200-2-3


The characteristic function for a contraction is a classical complete unitary invariant devised by Sz.-Nagy and Foias. Just as a contraction is related to the Szego kernel k(S) (z, w) = (1 - z (w) over tilde)(-1) for |z|, |w| < 1, by means of (1/k(S))(T,T*) >= 0, we consider an arbitrary open connected domain Omega in C-n, a complete Pick kernel k on Omega and a tuple T = (T-1, ..., T-n) of commuting bounded operators on a complex separable Hilbert space H such that (1/k)(T,T*) >= 0. For a complete Pick kernel the 1/k functional calculus makes sense in a beautiful way. It turns out that the model theory works very well and a characteristic function can be associated with T. Moreover, the characteristic function is then a complete unitary invariant for a suitable class of tuples T.

Item Type: Journal Article
Publication: Studia Mathematica
Publisher: Institute of Mathematics of the Polish Academy of Sciences
Additional Information: Copyright of this article belongs to Institute of Mathematics of the Polish Academy of Sciences.
Keywords: Complete Pick kernels; characteristic function; unitary invariance
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Date Deposited: 14 Dec 2010 10:10
Last Modified: 11 Oct 2018 14:48
URI: http://eprints.iisc.ac.in/id/eprint/34441

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