Semigroups of holomorphic isometries
type: Item
instance of: scholarly article
title: Semigroups of holomorphic isometries
zbMATH Open document ID: 0642.47035
DOI: 10.1016/0001-8708(87)90025-9
published in: Advances in Mathematics
publication date: 1987-01-01T00:00:00Z
review text: Let B be the open unit ball of a complex Hilbert space \({\mathcal H}\). Let \({\mathcal H}\oplus {\mathbb{C}}\) be the Hilbert space direct sum of \({\mathcal H}\) and \({\mathbb{C}}\), with inner product (,), and let \(\alpha\) be the continuous hermitian sesquilinear form defined by \(\alpha (p,q)=(Jp,q)\) where p,q\(\in {\mathcal H}\oplus {\mathbb{C}}\), J is the operator \(J=I\oplus (-1)\) and I is the identity on \({\mathcal H}\). \(G_ 0\) is the group of all continuous invertible linear operators in \({\mathcal H}\oplus {\mathbb{C}}\) leaving the form \(\alpha\) invariant. G is the semigroup G of all continuous linear operators in \({\mathcal H}\oplus {\mathbb{C}}\) leaving the form \(\alpha\) invariant. Aut B is the group of all holomorphic automorphisms of B. Iso B is the semigroup of all holomorphic maps \(B\to B\) which are isometries for the hyperbolic differential metric of B. The main result is the following: Theorem. There is a surjective homomorphism of G onto Iso B, mapping \(G_ 0\) onto Aut B, whose kernel is the center of G. The strongly continuous linear semigroup \(T:R_+\to G\) is characterized by use of their infinitesimal generator X. The Cauchy problem associated with the infinitesimal generator X of the semigroup T is presented.
reviewed by: Wu Liangsen
Mathematics Subject Classification ID: 47D03
zbMATH DE Number: 4046468
zbMATH Keywords: holomorphic isometries
zbMATH Keywords: Hilbert space direct sum
zbMATH Keywords: surjective homomorphism
zbMATH Keywords: strongly continuous linear semigroup
zbMATH Keywords: infinitesimal generator
zbMATH Keywords: Cauchy problem
author: Edoardo Vesentini
MaRDI profile type: Publication
full work available at URL: 0001-8708(87)90025-9
OpenAlex ID: W4210306565
cites work: Q3906660
cites work: Q3892778
cites work: Biholomorphic maps in Hilbert space have a fixed point
cites work: Q3244654
cites work: Q5186174
cites work: Q5526485
cites work: Semigroups of linear operators and applications to partial differential equations
cites work: Q5729425
cites work: Q3785139
cites work: Le groupe des automorphismes analytiques d'un domaine borné d'un espace de Banach complexe. Application aux domaines bornés symétriques
description: scientific article; zbMATH DE number 4046468