The holomorphic functional calculus approach to operator semigroups

In this article we construct a holomorphic functional calculus for operators of half-plane type and show how key facts of semigroup theory (Hille-Yosida and Gomilko-Shi-Feng generation theorems, Trotter-Kato approximation theorem, Euler approximation formula, Gearhart-Pruss theorem) can be elegantly obtained in this framework. Then we discuss the notions of bounded H-infinity-calculus and m-bounded calculus on half-planes and their relation to weak bounded variation conditions over vertical lines for powers of the resolvent. Finally we discuss the Hilbert space case, where semigroup generation is characterised by the operator having a strong m-bounded calculus on a half-plane.