Convergence of operator-semigroups associated with generalised elliptic forms

TL;DR

This paper advances the theory of operator semigroups linked to generalized elliptic forms, exploring perturbation and convergence properties, with applications to Laplace and Dirichlet-to-Neumann operators.

Contribution

It extends the analysis of j-elliptic forms by establishing new perturbation and convergence results for associated semigroups, including Schatten norm convergence.

Findings

Established convergence of semigroups in Schatten norms.

Applied results to Laplace and Dirichlet-to-Neumann operators.

Extended the theory of sectorial forms beyond closable cases.

Abstract

In a recent article, Arendt and ter Elst have shown that every sectorial form is in a natural way associated with the generator of an analytic strongly continuous semigroup, even if the form fails to be closable. As an intermediate step they have introduced so-called j-elliptic forms, which generalises the concept of elliptic forms in the sense of Lions. We push their analysis forward in that we discuss some perturbation and convergence results for semigroups associated with j-elliptic forms. In particular, we study convergence with respect to the trace norm or other Schatten norms. We apply our results to Laplace operators and Dirichlet-to-Neumann-type operators.