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