A Sharp Result on the Exponential Operator-Norm Decay of a Family Th (t) of Strongly Continuous Semigroups Uniformly in h

Given a family T_h(t) of s.c. semigroups on a Banach space X, we give a sharp result on the exponential decay in L https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072225/b8f6ffe5-10c1-4461-b09b-4c7a308c27a3/content/eq3905.tif"/> (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 s.c. semi-group. The obtained generalization plays a critical role in a variety of applications, from approximation theory to regularization theory.