Sharp estimates for large coupling convergence with applications to Dirichlet operators

Let H be a nonnegative selfadjoint operator, epsilon the closed quadratic form associated with H, and P a nonnegative quadratic form such that epsilon + P is closed and D(P) superset of D(H). For every beta > 0 let H-beta be the selfadjoint operator associated with epsilon + beta P. The pairs (H, P) satisfying L(H, P) := lim inf beta(beta ->infinity) parallel to(H-beta + 1)(-1) - lim(beta'->infinity) (H-beta' + 1)-1 parallel to < infinity are characterized. A sufficient condition for convergence of the operators (H-beta + 1)(-1) within a Schatten von Neumann class of finite order is derived. It is shown that L(H, P) = 1, if epsilon 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. (C) 2007 Elsevier Inc. All rights reserved.