Characterization of Dirichlet forms by their excessive and co-excessive elements (Q1266327)

Let \(( H,\langle\;,\;\rangle)\) be an ordered Hilbert space. A linear operator \(T\) on \(H\) is called a Dirichlet operator by the authors if there exists a strictly positive real number \(\alpha \) satisfying \(\langle Tx,x\rangle \geq \alpha \| x\| \). An element \(s\in H\) is called \(T\)-potential (resp. \(T\)-copotential) if \(\langle Ts,x\rangle \geq 0\) (resp. \(\langle Tx,s\rangle \geq 0\)) for all \(x\in H_{+}\). If any non-zero \(T\)-potential is a weak unit in \(H\), the operator is called elliptic. The authors prove a nice result that if \(S\) and \(T\) are Dirichlet operators and one of them is elliptic and their potentials and copotentials coincide then there exists a strictly positive real number \(\theta \) satisfying \(T=\theta S\). Moreover the authors show that two Dirichlet forms admitting the same domain of definition are equal if one of them is elliptic and their \(\alpha \)-excessive and \(\alpha \)-co-excessive elements coincide for any strictly positive \(\alpha \).