Sharp estimates for large coupling convergence with applications to Dirichlet operators

Abstract

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 H β be the selfadjoint operator associated with E + β P . The pairs ( H , P ) satisfying L ( H , P ) : = lim inf β → ∞ β ‖ ( H β + 1 ) −1 − lim β ′ → ∞ ( H β ′ + 1 ) −1 ‖ < ∞ are characterized. A sufficient condition for convergence of the operators ( H β + 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.