Spectral properties of Laplacians on an abstract Wiener space with a weighted Wiener measure
type: Item
instance of: scholarly article
title: Spectral properties of Laplacians on an abstract Wiener space with a weighted Wiener measure
zbMATH Open document ID: 0880.60064
author: Masanori Hino
published in: Journal of Functional Analysis
publication date: 1998-02-03T00:00:00Z
review text: Let \((B,H,\mu)\) be an abstract Wiener space and \(L\) be the Ornstein-Uhlenbeck operator over \((B,H,\mu)\). Shigekawa defined stochastic Sobolev spaces -- denoted in the sequel by \(D^p_s(K)\), where \(K\) is a given real Banach space -- as the completion of the class of ``smooth maps'' from \(B\) into \(K\) w.r.t. the following norms, \(|.|_{s,p,\kappa}:= |(\kappa- L)^{s/2}\cdot|_p\), \(\kappa>0\), \(s\in\mathbb{R}\), where \(|.|_p\) denotes the \(K\)-valued \(L_p\)-space \((1<p<+\infty)\). The class of \(m\)-forms over \((B,H,\mu)\) could then be introduced, so is the operator \(\square_m=(m-L)\), called the Hodge-Kodaira Laplacian over these forms. Aida and Shigekawa considered the abstract Wiener space \((B,H,\mu_F)\), \(\mu_F= e^{2F}d\mu/\int_B \exp(2F)d\mu\), where \(F\in D^2_1\) such that \(e^{|DF|^2_{H^*\in L^{-\infty}}}\) (Assumption A), and proved that its associated Laplacian \(L_F\) has only discrete spectra in \([0,1)\). The author investigates the ``spectral properties'' of the Hodge-Kodaira Laplacian over \((B,H,\mu_F)\), denoted by \(\square_{F,m}\), under weaker assumptions than (A), namely the following Assumption (I): \(e^{|DF|^2_{H^*\in L^{2+}}}= \bigcup_{p>2}L^p\) (from Section 1 to Section 4), Assumption (II): \(e^{|DF|^2_{H^*\in L^{4+}}}= \bigcup_{p>4} L^p\) (in the last Section 5). He shows in the first case that the kernels of \(\square_{F,m}\) and \(\square_m\) are identical (Theorem 4.6) and consequently \(\square_{F,m}\) has only discrete spectrum in \([0,1)\) for each \(m>0\) (Corollary 5.8), whereas in the second case the following theorem (Theorem 5.3) is valid: \[ \sigma_{\text{ess}}(\square_{F,m})= \{\sigma(\square_{F,m})\cup \sigma(\square_{F,m-1})\}+ 1. \] Note that under the weaker assumptions mentioned above the derivatives of the functional \(F\) may be unbounded, and the author uses here the Gross logarithmic inequalities in order to treat such unbounded perturbations of \(\square_{F,m}\).
Mathematics Subject Classification ID: 60H99
zbMATH DE Number: 1046935
zbMATH Keywords: spectral properties
zbMATH Keywords: Ornstein-Uhlenbeck operator
zbMATH Keywords: stochastic Sobolev spaces
zbMATH Keywords: Hodge-Kodaira Laplacian
zbMATH Keywords: logarithmic inequalities
zbMATH Keywords: unbounded perturbations
reviewed by: Xuan Loc Nguyen
MaRDI profile type: Publication
full work available at URL: jfan.1996.3072
OpenAlex ID: W2035570580
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DOI: 10.1006/JFAN.1996.3072
description: scientific article; zbMATH DE number 1046935