Holomorphic functions and the heat kernel measure on an infinite dimensional complex orthogonal group
type: Item
instance of: scholarly article
title: Holomorphic functions and the heat kernel measure on an infinite dimensional complex orthogonal group
zbMATH Open document ID: 0960.46026
DOI: 10.1023/A:1008626828889
author: Maria Gordina
published in: Potential Analysis
publication date: 2001-05-14T00:00:00Z
review text: The author studies Hilbert spaces of holomorphic functions over a particular infinite-dimensional (non-locally compact) noncommutative complex group; viz., the so-called Hilbert-Schmidt orthogonal group \(\text{SO}_{HS}\), embedded in the space of Hilbert-Schmidt operators on an infinite-dimensional Hilbert space. The class \(\text{so}_{HS}\) of skew symmetric Hilbert-Schmidt operators plays the role of a Lie algebra of \(\text{SO}_{HS}\). Let \(\text{SO}(\infty)\) denote the closure of the direct limit of a certain sequence of finite-dimensional subgroups of \(\text{SO}_{HS}\) in a Riemannian metric. To define the Hilbert spaces \({\mathcal H}L^2(\text{SO}_{HS}, \mu_t)\) and \({\mathcal H}L^2(\text{SO}(\infty))\) of holomorphic functions, the author first constructs the heat kernel measure \(\mu_t\) on \(\text{SO}_{HS}\). In the finite-dimensional case this measure has a density with respect to Haar measure, making it possible to apply techniques from partial differential equations. In the present infinite-dimensional case, however, stochastic differential equations in infinite dimensions are used to define \(\mu_t\) as the transition probability of a diffusion on \(\text{SO}_{HS}\). All holomorphic polynomials (in the matrix entries) on \(\text{SO}_{HS}\) are square integrable with respect to the heat kernel measure, and \({\mathcal H}L^2(\text{SO}_{HS}, \mu_t)\) is the closure in \(L^2(\text{SO}_{HS}, \mu_t)\) of the space of the holomorphic polynomials. \({\mathcal H}L^2(\text{SO}(\infty))\) denotes a certain Hilbert space of holomorphic functions on \(\text{SO}(\infty)\) with a direct limit type norm. It is proved that the embedding of the space of holomorphic polynomials into \({\mathcal H}L^2(\text{SO}(\infty))\) can be extended to an isometry from \({\mathcal H}L^2(\text{SO}_{HS}, \mu_t)\) into \({\mathcal H}L^2(\text{SO}(\infty))\). The main theorem is that an infinite-dimensional nonlinear analog of the Taylor expansion defines an isometry from \({\mathcal…
reviewed by: Klaus Dieter Bierstedt
Mathematics Subject Classification ID: 46E50
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 22E65
Mathematics Subject Classification ID: 58J65
Mathematics Subject Classification ID: 22E60
Mathematics Subject Classification ID: 46G25
Mathematics Subject Classification ID: 46G20
zbMATH DE Number: 1507482
zbMATH Keywords: infinite-dimensional group
zbMATH Keywords: holomorphic functions on an infinite-dimensional group
zbMATH Keywords: Hilbert spaces of holomorphic functions
zbMATH Keywords: Hilbert-Schmidt orthogonal group
zbMATH Keywords: Hilbert-Schmidt operators
zbMATH Keywords: Lie algebra
zbMATH Keywords: heat kernel measure
zbMATH Keywords: Haar measure
zbMATH Keywords: space of the holomorphic polynomials
zbMATH Keywords: embedding
zbMATH Keywords: Taylor expansion
zbMATH Keywords: Cameron-Martin subspace
MaRDI profile type: Publication
full work available at URL: a:1008626828889
OpenAlex ID: W2240668660
description: scientific article; zbMATH DE number 1507482