A SHARP RESULT ON THE EXPONENTIAL OPERATOR-NORM DECAY OF A FAMILY OF STRONGLY CONTINUOUS SEMIGROUPS

Given a family T(h)(t) of strongly continuous semigroups on a Banach space X, we give a sharp result on the exponential decay in L(X) of the family, which is uniform in the parameter h. 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.