Closability and resolvent of Dirichlet forms perturbed by jumps (Q1261222)

Let \(E\) be a locally compact Hausdorff space and \(m\) a \(\sigma\)-finite measure on the Borel sets of \(E\). Let \(\mathcal E\) be a symmetric Dirichlet form on \(L^ 2(E;m)\). The first main result of the paper is a necessary and sufficient condition for the closability on \(L^ 2(E;m)\) of perturbations of type \({\mathcal E}_ J = {\mathcal E}+J\), where \(J\) is a symmetric measure on the Borel sets of \(E\times E\). Furthermore, a very interesting formula for the resolvent of \({\mathcal E}_ J\) in terms of \(\mathcal E\) and \(J\) is derived, provided \(E\) is, in addition, a separable metric space, \(m\) is Radon, and \(\mathcal E\) is regular [in the sense of \textit{M. Fukushima}, Dirichlet forms and Markov processes (1980; Zbl 0422.31007)].