Sharp estimates for large coupling convergence with applications to Dirichlet operators

Let H   be a nonnegative selfadjoint operator, EE the closed quadratic form associated with H, and P   a nonnegative quadratic form such that E+PE+P is closed and D(P)⊃D(H)D(P)⊃D(H). For every β>0β>0 let HβHβ be the selfadjoint operator associated with E+βPE+βP. The pairs (H,P)(H,P) satisfyingL(H,P):=lim infβ→∞β‖(Hβ+1)−1−limβ′→∞(Hβ′+1)−1‖<∞ are characterized. A sufficient condition for convergence of the operators (Hβ+1)−1(Hβ+1)−1 within a Schatten–von Neumann class of finite order is derived. It is shown that L(H,P)=1L(H,P)=1, if EE 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)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.