Complete Pick positivity and unitary invariance

Angshuman Bhattacharya, and Tirthankar Bhattacharyya. "Complete Pick positivity and unitary invariance." Studia Mathematica 200.2 (2010): 149-162. <http://eudml.org/doc/285521>.

@article{AngshumanBhattacharya2010,
abstract = {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 $k_\{S\}(z,w) = (1-zw̅)^\{-1\}$ for |z|,|w| < 1, by means of $(1/k_\{S\})(T,T*) ≥ 0$, 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.},
author = {Angshuman Bhattacharya, Tirthankar Bhattacharyya},
journal = {Studia Mathematica},
keywords = {complete Pick kernel; commuting tuples; characteristic function; unitary invariance},
language = {eng},
number = {2},
pages = {149-162},
title = {Complete Pick positivity and unitary invariance},
url = {http://eudml.org/doc/285521},
volume = {200},
year = {2010},
}

TY - JOUR
AU - Angshuman Bhattacharya
AU - Tirthankar Bhattacharyya
TI - Complete Pick positivity and unitary invariance
JO - Studia Mathematica
PY - 2010
VL - 200
IS - 2
SP - 149
EP - 162
AB - 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 $k_{S}(z,w) = (1-zw̅)^{-1}$ for |z|,|w| < 1, by means of $(1/k_{S})(T,T*) ≥ 0$, 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.
LA - eng
KW - complete Pick kernel; commuting tuples; characteristic function; unitary invariance
UR - http://eudml.org/doc/285521
ER -