Algebraic structure on Dirichlet spaces (Q2508594)

The authors give a few equivalent conditions for a closed form \((\mathcal E,\mathcal F)\) on an \(L^2\)-space to be Markovian. It is well known that the semigroup generated by \(\mathcal E\) is sub-Markovian if and only if the unit/normal contractions operate on \(\mathcal E\). One of the equivalent properties reads as follows: the space \({\mathcal F}_b\) of essentially bounded functions is an algebra, and for any polynomial \(p\) with \(p(0)=0\) and for any \(u\in {\mathcal F}_b\) it follows that \(p(u)\in\mathcal F\) and \({\mathcal E}(p(u))\leq \| p'\| ^2_{\infty,u}{\mathcal E}(u)\). The authors also study the regular representation of Dirichlet spaces and the classification of Dirichlet subspaces.