On order isomorphisms intertwining semigroups for Dirichlet forms

TL;DR

This paper characterizes order isomorphisms that intertwine semigroups of Dirichlet forms, revealing their structure as compositions of transformations, quasi-homeomorphisms, and step functions under certain conditions.

Contribution

It provides a detailed description of the structure of order isomorphisms intertwining Dirichlet form semigroups, including the effects of absolute continuity conditions.

Findings

Unitary order isomorphisms are compositions of $h$-transformation and quasi-homeomorphism.

General order isomorphisms are compositions of $h$-transformation, quasi-homeomorphism, and step functions.

The characterization applies under specific conditions like absolute continuity.

Abstract

This paper is devoted to characterizing the so-called order isomorphisms intertwining the $L^2$-semigroups of two Dirichlet forms. We first show that every unitary order isomorphism intertwining semigroups is the composition of $h$-transformation and quasi-homeomorphism. In addition, under the absolute continuity condition on Dirichlet forms, every (not necessarily unitary) order isomorphism intertwining semigroups is the composition of $h$-transformation, quasi-homeomorphism, and multiplication by a certain step function.