Distance between reproducing kernel Hilbert spaces and geometry of finite sets in the unit ball

TL;DR

This paper introduces a new distance measure between reproducing kernel Hilbert spaces and their multiplier algebras, linking geometric properties of finite point sets in the unit ball to algebraic isomorphisms.

Contribution

It develops a novel Banach-Mazur type distance for RKHS and multiplier algebras, establishing a quantitative relationship between geometric configurations and algebraic isomorphisms in finite-dimensional settings.

Findings

Spaces are close iff their multiplier algebras are close

Closeness corresponds to point-sets being almost congruent under automorphisms

Quantitative estimates link geometric and algebraic proximity

Abstract

In this paper we study the relationships between a reproducing kernel Hilbert space, its multiplier algebra, and the geometry of the point set on which they live. We introduce a variant of the Banach-Mazur distance suited for measuring the distance between reproducing kernel Hilbert spaces, that quantifies how far two spaces are from being isometrically isomorphic as reproducing kernel Hilbert spaces. We introduce an analogous distance for multiplier algebras, that quantifies how far two algebras are from being completely isometrically isomorphic. We show that, in the setting of finite dimensional quotients of the Drury-Arveson space, two spaces are "close" to one another if and only if their multiplier algebras are "close", and that this happens if and only if the underlying point-sets are "almost congruent", meaning that one of the sets is very close to an image of the other under a biholomorphic automorphism of the unit ball. These equivalences are obtained as corollaries of quantitative estimates that we prove.