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_t,t?0} be a subordinator with Laplace exponent φ that is independent of X. The processes X^φ?{X_St,t?0} and are called the subordinate processes of X and X^D, respectively. Under some mild conditions, we show that, if {-μ_n,n?1} and {-λ_n,n?1} denote the eigenvalues of the generators of the subprocess of X^φ killed upon leaving D and of the process X^D respectively, then