Heat kernel analysis of Cameron-Martin subgroup for infinite dimensional groups
type: Item
instance of: scholarly article
title: Heat kernel analysis of Cameron-Martin subgroup for infinite dimensional groups
zbMATH Open document ID: 1041.46027
author: Maria Gordina
published in: Journal of Functional Analysis
publication date: 2000-08-28T00:00:00Z
review text: In \textit{M. Gordina} [Potential Anal. 12, No. 4, 325--357 (2000, Zbl 0960.46026)], the author had constructed the heat kernel measure \(\mu_t\) on the Hilbert-Schmidt orthogonal group \(\text{SO}_{\text{HS}}\), embedded in the space of Hilbert-Schmidt operators on an infinite-dimensional Hilbert space; then she had studied Hilbert spaces of holomorphic functions over \(\text{SO}_{\text{HS}}\). At the end of the article, she had claimed that her methods can be applied in more general situations. The aim of the present article is to verify this claim: The author generalizes her previous work to a setting which also includes examples like the Hilbert-Schmidt complex symplectic group \(\text{Sp}_{\text{HS}}\) or a group \(G\) of diagonal (infinite) complex matrices \(\text{diag}(1+ a_1,\dots, 1+ a_i,\dots)\), where \(\sum^\infty_{i=1}| a_i|^2< \infty\), \(a_i\neq 1\). From the author's abstract: ``The heat kernel measure \(\mu_t\) is constructed on \(\text{GL}(H)\), the group of invertible operators on a complex Hilbert space \(H\). This measure is determined by an infinite-dimensional Lie algebra \({\mathfrak g}\) and a Hermitian inner product on it. The Cameron-Martin subgroup \(G_{\text{CM}}\) is defined and its properties are discussed. In particular, there is an isometry from the \(L^2_{\mu_t}\)-closure of holomorphic polynomials into a space \({\mathcal H}^t(G_{\text{CM}})\) of functions holomorphic on \(G_{\text{CM}}\). This means that any element from this \(L^2_{\mu_t}\)-closure of holomorphic polynomials has a version holomorphic on \(G_{\text{CM}}\). In addition, there is an isometry from \({\mathcal H}^t(G_{\text{CM}})\) into a Hilbert space associated with the tensor algebra over \({\mathfrak g}\). The latter isometry is an infinite-dimensional analog of the Taylor expansion.''
reviewed by: Klaus Dieter Bierstedt
Mathematics Subject Classification ID: 46E50
Mathematics Subject Classification ID: 60B15
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 22E65
Mathematics Subject Classification ID: 46N20
Mathematics Subject Classification ID: 60H07
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 46G20
Mathematics Subject Classification ID: 22E60
zbMATH DE Number: 1423459
zbMATH Keywords: heat kernel measure
zbMATH Keywords: holomorphic function
zbMATH Keywords: infinite-dimensional group
zbMATH Keywords: infinite-dimensional Lie algebra
zbMATH Keywords: stochastic differential equation
zbMATH Keywords: Cameron-Martin subgroup
zbMATH Keywords: tensor algebra
zbMATH Keywords: Hilbert-Schmidt orthogonal group
zbMATH Keywords: Hilbert-Schmidt symplectic group
MaRDI profile type: Publication
full work available at URL: jfan.1999.3505
OpenAlex ID: W2089158521
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DOI: 10.1006/JFAN.1999.3505
description: scientific article; zbMATH DE number 1423459