Two-sided eigenvalue estimates for subordinate processes in domains

Let X={Xt ,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 ={ St ,t 0} be a subordinator with Laplace exponent that is independent of X. The processes X := {XSt ,t 0} and (X D ) := {X Dt ,t 0} are called the subordinate processes of X and X D , respectively. Under some mild conditions, we show that, if {−n ,n 1} and {−n ,n 1} denote the eigenvalues of the generators of the subprocess of X killed upon leaving D and of the process X D respectively, then

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