Continuity of eigenvalues of 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. Let S(ⁿ) = {S_t( ⁿ), 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 [InlineMediaObject not available: see fulltext.] and [InlineMediaObject not available: see fulltext.] and their subprocesses [InlineMediaObject not available: see fulltext.] and X^{φn, D} killed upon leaving D. Suppose that the spectra of the semigroups of [InlineMediaObject not available: see fulltext.] and X^{φ, D} are all discrete, with [InlineMediaObject not available: see fulltext.] being the eigenvalues of the generator of [InlineMediaObject not available: see fulltext.] and [InlineMediaObject not available: see fulltext.] 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 [InlineMediaObject not available: see fulltext.].