Complete Pick Positivity and Unitary Invariance

TL;DR

This paper extends the concept of characteristic functions as complete unitary invariants from classical contraction theory to tuples of commuting operators on domains with complete Nevanlinna-Pick kernels, broadening the scope of model theory.

Contribution

It introduces a new framework for characteristic functions in multivariable operator theory on domains with complete Pick kernels, establishing their role as complete invariants.

Findings

Characteristic functions serve as complete unitary invariants for certain operator tuples.

The $1/k$ functional calculus is well-defined and effective in this setting.

Model theory extends naturally to these operator tuples with the new invariants.

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 $k_S(z,w) = (1 - z\ow)^{-1}$ 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 $T$.