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

Danny Ofek, Satish K. Pandey, Orr Shalit

TL;DR

This work analyzes how the geometry of a finite point set $X$ in the unit ball dictates the structure of the associated RKHS $\mathcal{H}_X$ and its multiplier algebra $\mathcal{M}_X$, and vice versa. It introduces a reproducing-kernel Banach-Mazur distance $\rho_{RK}$ and a multiplier Banach-Mazur distance $\rho_M$, along with an automorphism-invariant Hausdorff distance, and proves that small geometric distance corresponds to small $\rho_{RK}$ and $\rho_M$ for finite quotients of the Drury–Arveson space; conversely, small $\rho_{RK}$ and $\rho_M$ force the point sets to be nearly congruent under a biholomorphic automorphism of the ball. The core result establishes an equivalence among $\tilde{\rho}_s(X,Y)$, $\rho_{RK}(\mathcal{H}_X,\mathcal{H}_Y)$, and $\rho_M(\mathcal{M}_X,\mathcal{M}_Y)$ for finite $X,Y$, with explicit quantitative bounds linking these invariants. These findings bridge finite-point geometry in the ball with operator-algebraic invariants, yielding corollaries about multiplier discrepancy and geometric near-congruence that inform the isomorphism problem for complete Pick spaces.

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.

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

TL;DR

This work analyzes how the geometry of a finite point set in the unit ball dictates the structure of the associated RKHS and its multiplier algebra , and vice versa. It introduces a reproducing-kernel Banach-Mazur distance and a multiplier Banach-Mazur distance , along with an automorphism-invariant Hausdorff distance, and proves that small geometric distance corresponds to small and for finite quotients of the Drury–Arveson space; conversely, small and force the point sets to be nearly congruent under a biholomorphic automorphism of the ball. The core result establishes an equivalence among , , and for finite , with explicit quantitative bounds linking these invariants. These findings bridge finite-point geometry in the ball with operator-algebraic invariants, yielding corollaries about multiplier discrepancy and geometric near-congruence that inform the isomorphism problem for complete Pick spaces.

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.

Paper Structure

This paper contains 13 sections, 16 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Background on the isomorphism problem
  3. What this paper is about
  4. Main results
  5. Preliminaries
  6. Notions of isomorphism for reproducing kernel Hilbert spaces
  7. Notions of isomorphism for multiplier algebras
  8. Some useful facts about isomorphisms of RKHSs and multiplier algebras on finite subsets of the ball
  9. The pseudohyperbolic metric
  10. Distances between subsets of the ball
  11. Distances for reproducing kernel Hilbert spaces and multiplier algebras
  12. Structure of spaces and algebras versus geometry of point sets
  13. Examples

Key Result

Lemma 3.2

If $X = \{x_1, \ldots, x_n\}$ and $Y = \{y_1, \ldots, y_n\}$ are subsets of a metric space $(M,\rho)$, then $\rho_{H}(X,Y)\leq \rho_s(X,Y)$.

Theorems & Definitions (44)

  • Definition 2.1
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Corollary 3.4
  • Example 3.5
  • Definition 3.6
  • Lemma 3.7
  • ...and 34 more