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

Danny Ofek, Satish K. Pandey, Orr Moshe Shalit

Research output: Contribution to journalArticlepeer-review

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 one of the underlying point sets is 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.

Original languageEnglish
Article number125140
JournalJournal of Mathematical Analysis and Applications
Volume500
Issue number2
DOIs
StatePublished - 15 Aug 2021
Externally publishedYes

Keywords

  • Multiplier Banach-Mazur distance
  • Multiplier algebras
  • Reproducing kernel Banach-Mazur distance
  • Reproducing kernel Hilbert spaces

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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