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Studia Mathematica

2010 | 200 | 2 | 149-162

Complete Pick positivity and unitary invariance

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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.

149-162

wydano
2010

Twórcy

• Indian Institute of Science, Bangalore 560012, India
• Indian Institute of Science, Bangalore 560012, India