Essential spectrum and Feller type properties

TL;DR

This paper characterizes when regular semi-Dirichlet forms exhibit a new weak Feller property, linking potential theory and probability, and derives new spectral decomposition results applicable to Cheeger forms on RCD* spaces.

Contribution

It introduces a novel weak Feller property for semi-Dirichlet forms and provides new spectral decomposition principles, extending existing theories in symmetric cases.

Findings

Characterization of weak Feller property involving potential and probabilistic aspects

New spectral decomposition principle for symmetric Dirichlet forms

Applicability to Cheeger forms on RCD* spaces

Abstract

We give necessary and sufficient conditions for a regular semi-Dirichlet form to enjoy a new Feller type property, which we call \emph{weak Feller property}. Our characterization involves potential theoretic as well as probabilistic aspects and seems to be new even in the symmetric case. As a consequence, in the symmetric case, we obtain a new variant of a decomposition principle of the essential spectrum for (the self-adjoint operators induced by) regular symmetric Dirichlet forms and a Persson type theorem, which applies e.g. to Cheeger forms on $\mathsf{RCD^*}$ spaces.