Abstract characteristic function

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−zw)^-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 kernel k on Ω so that 1/k is a polynomial and a tuple T = (T₁, T₂, . . . , T_n) of commuting bounded operators on a complex separable Hilbert space H such that (1/k)(T, T*) ≥ 0. Under some standard assumptions on k, it turns out that whether a characteristic function can be associated with T or not depends not only on T, but also on the kernel k. We give a necessary and sufficient condition. When this condition is satisfied, a functional model can be constructed. Moreover, the characteristic function then is a complete unitary invariant for a suitable class of tuples T.