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 d-dimensional 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 non-negative self-adjoint
operators whose n-th 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. self-simiar 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 sub-Gaussian 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 Lapla-
cian 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.