Spectral volume and surface measures via the Dixmier trace for local symmetric Dirichlet spaces with Weyl type eigenvalue asymptotics
Spectral volume and surface measures via the Dixmier trace for local symmetric Dirichlet spaces with Weyl type eigenvalue asymptotics The purpose of this talk is to present the author's recent results of on an operator theoretic way of looking atWeyl type Laplacian eigenvalue asymptotics for local symmetric Dirichlet spaces.For the Laplacian on a ddimensional Riemannian manifoldM, Connes' trace theorem implies that the linear functional coincides with (a constant multiple of) the integral with respect to the Riemannian volume measure of M, which could be considered as an operator theoretic paraphrase of Weyl's Laplacian eigenvalue asymptotics. Here denotes a Dixmier trace which is a trace functional de_ned on a certain ideal of compact operators on a Hilbert space and is meaningful e.g. for compact nonnegative selfadjoint operators whose nth largest eigenvalue is comparable to 1/n.The first main result of this talk is an extension of this fact in the framework of a general regular symmetric Dirichlet space satisfying Weyl type asymptotics for the trace of its associated heat semigroup, which was proved for Laplacians on p.c.f. selfsimiar sets by Kigami and Lapidus in 2001 under a rather strong assumption.
Moreover, as the second main result of this talk it is also shown that, given a local regular symmetric Dirichlet space with a subGaussian heat kernel upper bound and a (sufficiently regular) closed subset S, a “spectral surface measure" on S can be obtained through a similar linear functional involving the Laplacian with Dirichlet boundary condition on S. In principle, corresponds to the second order term for the eigenvalue asymptotics of this Dirichlet Laplacian, and when the second order term is explicitly known it is possible to identify For example, in the case of the usual Laplacian on Rd and a Lipschitz hypersurface S,is a constant multiple of the usual surface measure on S.
Location:  
Speaker(s):  Naotaka Kauine (Kobe University Japan)

Date:  Monday, February 2, 2015  15:45 