Complete pick positivity and unitary invariance

Abstract

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 k_S(z,w)=(1−zϖ)⁻¹ for |z|,|w|<1, by means of (1/kS)(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,…,T_n) 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.