Spectral properties of Laplacians on an abstract Wiener space with a weighted Wiener measure
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 \(\square_{F,m}\).
scientific article; zbMATH DE number 1046935