Continuity of eigenvalues of subordinate processes in domains

Abstract

Let X={X _t ,t≥0} be a symmetric Markov process in a state space E and D an open set of E. Let S⁽ⁿ⁾={S⁽ⁿ⁾ _t , t ≥ 0} be a subordinator with Laplace exponent ϕ _n and S={S _t ,t≥0} a subordinator with Laplace exponent ϕ. Suppose that X is independent of S and S⁽ⁿ⁾. In this paper we consider the subordinate processes and and their subprocesses and X^ϕ,D killed upon leaving D. Suppose that the spectra of the semigroups of and X^ϕ,D are all discrete, with being the eigenvalues of the generator of and being the eigenvalues of the generator of X^ϕ,D. We show that, if lim _{n →∞}ϕ _n (λ)=ϕ(λ) for every λ>0, then