Semistable Operators and Singularly Perturbed Differential Equations

where A and B are bounded linear operators on a Banach spaces X. w x Campbell studied the limit in his interesting monograph 1 in the case that A and B are finite matrices, and showed that the limit exists for an arbitrary B provided that A is semistable. Apart from the intrinsic interest in extending this problem to operators, there seems to be a need for a new proof that would be more transparent than Campbell’s original proof. The w x argument in 1 uses matrix specific techniques, such as the Jordan form, and resorts to numerical range estimates and a block version of the Gershgorin theorem to obtain a localization of eigenvalues of the parameter dependent matrices. Our proof relies mainly on the upper semicontinuity of the spectrum and on a uniform perturbation result for resolvents, and is simpler even when applied to matrices. The main theorem is then applied to the differential equation