Complete Pick Positivity and Unitary Invariance

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 $k_S(z,w)=(1-z\text{ow}{)}^{-1}$ for $\mathrm{\mid }z\mathrm{\mid },\mathrm{\mid }w\mathrm{\mid }<1$, by means of $(1\mathrm{/}{k}_S)(T,T^{\ast })\ge 0$, we consider an arbitrary open connected domain $\mathrm{\Omega }$ in ${\text{BC}}^n$, a complete Nevanilinna-Pick kernel $k$ on $\mathrm{\Omega }$ and a tuple $T = (T_1, …, T_n)$ of commuting bounded operators on a complex separable Hilbert space $\text{clh}$ such that $(1\mathrm{/}k)(T,T^{\ast })\ge 0$. For a complete Pick kernel the $1\mathrm{/}k$ 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 $T$.