Two-sided eigenvalue estimates for subordinate processes in domains

Let X = {X-t, t >= 0) be a symmetric Markov process in a state space E and D an open set of E. Denote by X D the subprocess of X killed upon leaving D. Let S = ( S, 7 t >= 0) be a subordinator with Laplace exponent (P that is independent of X. The processes X-phi := {X-St, t >= 0) and (X-D)(phi) := (X-St(D), t >= 0} are called the subordinate processes of X and XD, respectively. Under S, some mild conditions, we show that, if it {-mu(n), n >= 1} and (-lambda(n), n >= 1} denote the eigenvalues of the generators of the subprocess of X-phi killed upon leaving D and of the process X-D respectively, then mu(n) <= phi(lambda(n)) for every n >= 1. We further show that, when X is a spherically symmetric a-stable process in R-d with a E (0, 21 and D c Rd is a bounded domain satisfying the exterior cone condition, there is a constant c = c(D) > 0 such that c phi(lambda(n))<=mu(n)<=phi(lambda(n)) fore every n >= 1.