Complex Hyperbolic Geometry and Hilbert Spaces with the Complete Pick Property

TL;DR

This paper explores the connection between the geometry of complex hyperbolic spaces and the function theory of certain Hilbert spaces with the complete Pick property, revealing how geometric embeddings relate to function-theoretic properties.

Contribution

It establishes an isometric embedding of spaces with the complete Pick property into complex hyperbolic space, linking geometric and function-theoretic aspects.

Findings

Isometric embedding of $X$ into $ ext{CH}^n$ via $ ext{H}$

Relationship between the geometry of $ ext{Φ}(X)$ and the multiplier algebra

Characterization of the complete Pick property in geometric terms

Abstract

Suppose $H$ is a finite dimensional reproducing kernel Hilbert space of functions on $X.$ If $H$ has the complete Pick property then there is an isometric map, $\Phi,$ from $X,$ with the metric induced by $H,$ into complex hyperbolic space, $\mathbb{CH}^{n},$ with its pseudohyperbolic metric. We investigate the relationships between the geometry of $\Phi(X)$ and the function theory of $H$ and its multiplier algebra.