On uniqueness problem for local Dirichlet forms (Q1355278)

Let \(X\) be a locally compact separable complete metric space and \(({\mathcal E}, {\mathcal F})\) be a Dirichlet space satisfying a strong locality property. The authors consider the Riemannian pseudometric \(\rho\) on \(X\) associated with \(({\mathcal E}, {\mathcal F})\) and they assume that (1) \(\rho\) is a metric defining the original topology on \(X\). Under the assumption (1) the authors prove the uniqueness of the extension in Silverstein sense of \(({\mathcal E}, {\mathcal F})\). Let now \(X\) be a smooth manifold and \(A\) be the selfadjoint operator relative to \(({\mathcal E}, {\mathcal F})\) and assume that the domain of \(A\) contains \(C_0^\infty (X)\). Denote by \(A\uparrow C_0^\infty (X)\) the restriction of \(A\) to \(C_0^\infty (X)\). The authors prove that if \(A\) is hypoelliptic and under the assumption (1) then \(A\uparrow C_0^\infty (X)\) is essentially selfadjoint.