Locality and Domination of Semigroups

TL;DR

This paper characterizes all semigroups on L^2(Ω) that are bounded between Dirichlet and Neumann semigroups, removing the usual locality assumption using Beurling-Deny and Lejan formulas.

Contribution

It provides a complete characterization of semigroups between Dirichlet and Neumann types without assuming locality, expanding understanding of semigroup structures.

Findings

Characterization of semigroups between Dirichlet and Neumann semigroups.

Removal of locality assumption in semigroup analysis.

Application of Beurling-Deny and Lejan formulas.

Abstract

We characterize all semigroups $(T(t))_{t\geq0}$ on $L^2(\Omega)$ sandwiched between Dirichlet and Neumann ones, i.e.: \begin{equation*}\label{eq:san} e^{t\Delta_D}\leq T(t)\leq e^{t\Delta_N}\quad,\text{for all }t\geq0 \end{equation*} in the positive operators sense. The proof uses the well-known Beurling-Deny and Lejan formula to drop the locality assumption made usually on the form associated with $(T(t))_{t\geq 0}$.