On Order Isomorphisms Intertwining Semigroups for Dirichlet Forms

This paper is devoted to characterizing so-called order isomorphisms intertwining the $$L^2$$ -semigroups of two quasi-regular Dirichlet forms. We first show that every unitary isomorphism semigroups composition h-transformation and quasi-homeomorphism. In addition, under assumption underlying spaces admit irreducible decompositions for forms, (not necessarily unitary) can be expressed as h-transformation, quasi-homeomorphism multiplication by a certain step function.