Complete Pick Positivity and Unitary Invariance

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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 kₛ(z,w) = (1 - z)⁻¹ for |z|, |w| < 1, by means of (1/kₛ)(T,T*) ≥ 0, we consider an arbitrary open connected domain Ω in ⁿ, a complete Nevanilinna-Pick kernel k on Ω and a tuple T = (T₁,..., Tₙ) of commuting bounded operators on a complex separable Hilbert space 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 then is a complete unitary invariant for a suitable class of tuples T.