Ž . Journal of Mathematical Analysis and Applications 231, 446458 1999 Article ID jmaa.1998.6235, available online at http:www.idealibrary.com on Semistable Operators and Singularly Perturbed Differential Equations J. J. Koliha and Trung Dinh Tran* Department of Mathematics and Statistics, Uni  ersity of Melbourne, Park  ille 3052, Australia Submitted by William F. Ames Received October 21, 1997 1. INTRODUCTION AND PRELIMINARIES The main aim of this paper is to investigate the existence of the limit lim exp sA  B , Ž . s where A and B are bounded linear operators on a Banach spaces 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  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 parame- ter dependent matrices. Our proof relies mainly on the upper semicontinu- ity 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 dx t A  Ž. Ž .   B  xt ,   0,  , t  0, Ž . Ž. Ž . 0 ž / dt  x 0  u , Ž. *Supported by an Australian Commonwealth Postgraduate Award. 446 0022-247X99 $30.00 Copyright  1999 by Academic Press All rights of reproduction in any form reserved.