From Resolvent Estimates to Semigroup Bounds

In Chap. 10 we saw a concrete example of how to get resolvent bounds from semigroup bounds. Naturally, one can go in the opposite direction and in this chapter we discuss some abstract results of that type, including the Hille–Yoshida and Gearhardt–Pruss–Hwang–Greiner theorems. As for the latter, we also give a result of Helffer and the author that provides a more precise bound on the semigroup.