Difference estimates for continuous and discrete operator semigroups

Abstract

For suitable bounded operator semigroups (etA)t≥0 in a Banach space, we characterize the estimate ‖AetA‖≤c/F(t) for large t, where F is a function satisfying a sublinear growth condition. The characterizations are by holomorphy estimates on the semigroup, and by estimates on powers of the resolvent. We give similar characterizations of the difference estimate ‖Tn−Tn+1‖≤c/F(n) for a power-bounded linear operator T, when F(n) grows faster than n1/2 for large n.