Complete Pick positivity and unitary invariance
Tom 200 / 2010
Streszczenie
The characteristic function for a contraction is a classical complete unitary invariant devised by Sz.-Nagy and Foiaş. Just as a contraction is related to the Szegö kernel $k_S(z,w) = (1 - z\overline w)^{-1}$ for $|z|,|w| < 1$, by means of $(1/k_S)(T,T^*) \ge 0$, we consider an arbitrary open connected domain $\mit\Omega$ in ${\mathbb C}^n$, a complete Pick kernel $k$ on $\mit\Omega$ and a tuple $T = (T_1, \ldots ,T_n)$ of commuting bounded operators on a complex separable Hilbert space $\cal H$ 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 is then a complete unitary invariant for a suitable class of tuples $T$.
