The characteristic function for a contraction is a classical complete unitary invariant devised by Sz.-Nagy and Foiaş. Just as a contraction is related to the Szegö kernel for |z|,|w| < 1, by means of , we consider an arbitrary open connected domain Ω in ℂⁿ, a complete Pick kernel k on Ω and a tuple T = (T₁, ..., Tₙ) 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm200-2-3,
author = {Angshuman Bhattacharya and Tirthankar Bhattacharyya},
title = {Complete Pick positivity and unitary invariance},
journal = {Studia Mathematica},
volume = {196},
year = {2010},
pages = {149-162},
zbl = {1215.47010},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm200-2-3}
}
Angshuman Bhattacharya; Tirthankar Bhattacharyya. Complete Pick positivity and unitary invariance. Studia Mathematica, Tome 196 (2010) pp. 149-162. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm200-2-3/