Automorphism invariant inner product in Hilbert spaces of holomorphic functions on the unit ball of \(\mathbb{C}^ n\)
type: Item
instance of: scholarly article
title: Automorphism invariant inner product in Hilbert spaces of holomorphic functions on the unit ball of \(\mathbb{C}^ n\)
zbMATH Open document ID: 0877.46012
author: Yanyan Li
published in: Osaka Journal of Mathematics
publication date: 1997-11-27T00:00:00Z
review text: Let \(B\) be the open unit ball in complex \(n\)-space \(\mathbb{C}^n\) and \(\Aut(B)\) denote the space of holomorphic automorphisms of \(B\) defined by the Möbius transformations of \(B\) onto itself. A result of \textit{K. Zhu} [Trans. Am. Math. Soc. 323, No. 2, 823-842 (1991; Zbl 0739.46009)] says that there is a unique Hilbert space \(H\) of holomorphic functions on \(B\) which is \(\Aut(B)\) invariant. In the present paper, the author gives two explicit integral formulas for the \(\Aut(B)\) invariant inner product in \(H\). These formulas are first derived on the space \(P(B)\) of holomorphic polynomial functions from \(B\) to \(C\) and then extended to the Hilbert space completion \(H\) of \(P(B)\).
reviewed by: James V. Whittaker
Mathematics Subject Classification ID: 46E20
Mathematics Subject Classification ID: 32A10
Mathematics Subject Classification ID: 32N15
Mathematics Subject Classification ID: 46E10
zbMATH DE Number: 880170
zbMATH Keywords: Möbius invariant Hilbert space
zbMATH Keywords: space of holomorphic automorphisms
zbMATH Keywords: Möbius transformations
zbMATH Keywords: integral formulas
zbMATH Keywords: invariant inner product
zbMATH Keywords: holomorphic polynomial functions
zbMATH Keywords: Hilbert space completion
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 880170