Sharp estimates for large coupling convergence with applications to Dirichlet operators

Let H be a nonnegative selfadjoint operator, E the closed quadratic form associated with H, and P a nonnegative quadratic form such that E + P is closed and D(P) ⊃ D(H). For every β >0 let Hβ be the selfadjoint operator associated with E +βP. The pairs (H,P) satisfying L(H,P) := lim inf β→∞ β (Hβ +1)−1 − lim β →∞ (Hβ + 1)−1 <∞ are characterized. A sufficient condition for convergence of the operators (Hβ + 1)−1 within a Schatten– von Neumann class of finite order is derived. It is shown that L(H,P) = 1, if E is a regular conservative Dirichlet form with the strong local property and P the killing form corresponding to the equilibrium measure of a closed set with finite capacity and nonempty interior. An example is given where L(H,P) is finite, H is a regular Dirichlet operator and P the killing form corresponding to a measure which has infinite mass and a support with infinite capacity. © 2007 Elsevier Inc. All rights reserved.