Complete Pick positivity and unitary invariance

• Autorzy: Angshuman Bhattacharya , Tirthankar Bhattacharyya
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 47A48: Operator colligations (= nodes), vessels, linear systems, characteristic functions, realizations, etc.
• 47A13: Several-variable operator theory (spectral, Fredholm, etc.)
• 32A70: Functional analysis techniques \SeeMainly{}

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2010
• Tom: 200
• Numer: 2
• Strony: 149-162

Daty

wydano

2010

Twórcy

autor

Angshuman Bhattacharya

• Indian Institute of Science, Bangalore 560012, India

autor

Tirthankar Bhattacharyya

• Indian Institute of Science, Bangalore 560012, India

Identyfikatory

DOI

10.4064/sm200-2-3