A sharp result on the exponential operator-norm decay of a family of strongly continuous semigroups

Given a familyT _h (t) of strongly continuous semigroups on a Banach spaceX, we give a sharp result on the exponential decay inL(X) of the family, which is uniform in the parameterh. Sharpness is shown by means of counterexamples. This result generalizes a well-known and very useful criterion of stability of a single strongly continuous semigroup. The obtained generalization plays a critical role in a variety of applications, from approximation theory to regularization theory. Research partially supported by the National Science Foundation under Grant NSF-DMS-8902811-01.