# On the decomposition principle and a Persson type theorem for general regular Dirichlet forms

## TL;DR

This paper introduces a decomposition principle for regular Dirichlet forms under local compactness, leading to a Persson type theorem applicable to various operators including Laplace-Beltrami and stable processes.

## Contribution

It establishes a new decomposition principle and derives a Persson type theorem for general regular Dirichlet forms, broadening the scope of spectral analysis tools.

## Key findings

- Decomposition principle valid for regular Dirichlet forms with spatial local compactness
- Derived a Persson type theorem for these forms
- Applicable to Laplace-Beltrami operators and stable processes

## Abstract

We present a decomposition principle for general regular Dirichlet forms satisfying a spatial local compactness condition. We use the decomposition principle to derive a Persson type theorem for the corresponding Dirichlet forms. In particular our setting covers Laplace-Beltrami operators on Riemannian manifolds, and Dirichlet forms associated to $\alpha$-stable processes in Euclidean space.