Invariants Associated with Complete Nevanlinna-Pick Spaces

Abstract

The broad aim of this thesis is to study certain tuples of bounded operators that satisfy an operator inequality arising from a complete Nevanlinna-Pick kernel. Specifically, we focus on two invariants of these operator tuples: the characteristic function and the curvature invariant.

We extend the theory of Sz.-Nagy and Foias about the characteristic function of a contraction to a commuting tuple $(T_{1}, \dots, T_{d})$ of bounded operators satisfying the natural positivity condition of $1/k$-contractivity for a unitarily invariant complete Nevanlinna-Pick kernel. The characteristic function is a multiplier from $H_k \otimes \cE$ to $H_k \otimes \cF$, {\em factoring} a certain positive operator, for suitable Hilbert spaces $\cE$ and $\cF$ depending on the $d$-tuple $(T_{1}, \dots, T_{d})$. Surprisingly, there is a converse, which roughly says that if a kernel $k$ {\em admits} a characteristic function, then it has to be a unitarily invariant complete Nevanlinna-Pick kernel. The characterization explains, among other things, why in the literature, an analogue of the characteristic function for a Bergman contraction ($1/k$-contraction where $k$ is the Bergman kernel of the unit ball), when viewed as a multiplier between two vector-valued reproducing kernel Hilbert spaces, requires a different (vector-valued) reproducing kernel Hilbert space as the domain. In fact, in this case, the reproducing kernel Hilbert space, which has served as the domain of the multiplication operator, is the vector-valued Drury-Arveson space.

So, what can be said if the $d$-tuple $(T_{1}, \dots, T_{d})$ is $1/k$-contractive when $k$ is a unitarily invariant kernel but does not have the complete Nevanlinna-Pick property? We present a unified framework for deriving characteristic functions for kernels that allow a complete Nevanlinna-Pick factor. Notably, our approach not only encapsulates all previously documented cases but also achieves a remarkable level of generalization, thereby expanding the concept of the characteristic function substantially. We also provide an explanation for the prominence of the Drury-Arveson kernel in all previously established results by showing that the Drury-Arveson kernel was the natural choice inherently suitable for those situations.

We associate with a $1/k$-contraction its curvature invariant. The instrument that makes this possible is the characteristic function. We present an asymptotic formula for the curvature invariant. In the special case when the $1/k$-contraction is pure, we provide a notably simpler formula, revealing that, in this instance, the curvature invariant is an integer. We further investigate its connection with an algebraic invariant known as fiber dimension. Moreover, we obtain a refined and simplified asymptotic formula for the curvature invariant of the $1/k$-contraction, specifically when its characteristic function is a polynomial.