Deformations of Reproducing Kernel Hilbert Spaces on Homogeneous Subvarieties of the Unit Ball

TL;DR

This paper investigates the relationships between subvarieties of the unit ball, their associated reproducing kernel Hilbert spaces, and multiplier algebras, establishing conditions under which these structures are almost isometrically or automorphically equivalent.

Contribution

It demonstrates that almost isometric isomorphisms of RKHSs imply similar isomorphisms of multiplier algebras and geometric equivalences of the underlying varieties, especially for homogeneous varieties.

Findings

Almost isometric isomorphisms of RKHSs imply isomorphisms of multiplier algebras.

Multiplier algebra isomorphisms imply geometric equivalences of varieties.

Results apply to tractable homogeneous varieties under unitary transformations.

Abstract

We study the relationships between a subvariety of the open unit ball in the complex $d$-dimensional space $\mathbb{C}^{d}$, the reproducing kernel Hilbert space (RKHS) obtained by restricting the Drury-Arveson space to the variety, and its multiplier algebra. We show that if two such RKHSs are almost isometrically isomorphic as RKHSs, their multiplier algebras are likewise almost completely isometrically isomorphic as multiplier algebras. In such cases, the underlying varieties are almost automorphically equivalent. For tractable homogeneous varieties, we further show that if the corresponding multiplier algebras are almost completely isometrically isomorphic as multiplier algebras or one variety is almost the image of the other under a unitary transformation, then the associated RKHSs are almost isometrically isomorphic as RKHSs.