# On the construction and convergence of traces of forms

## TL;DR

This paper introduces a new method for constructing traces of quadratic forms using monotone convergence and form decomposition, with applications to Dirichlet forms and their convergence properties.

## Contribution

It develops a novel approach to trace construction for quadratic forms and links Mosco convergence of forms to their traces, enhancing understanding of form approximation.

## Key findings

- Trace can be explicitly described in various cases.
- Mosco convergence of Dirichlet forms implies convergence of their traces.
- Asymptotic compactness of forms leads to compactness of traces.

## Abstract

We elaborate a new method for constructing traces of quadratic forms in the framework of Hilbert and Dirichlet spaces. Our method relies on monotone convergence of quadratic forms and the canonical decomposition into regular and singular part. We give various situations where the trace can be described more explicitly and compute it for some illustrating examples. We then show that Mosco convergence of Dirichlet forms implies Mosco convergence of a subsequence of their approximating traces and that asymptotic compactness of Dirichlet forms yields asymptotic compactness of their traces.