Functional calculus for bisectorial operators and applications to linear and non-linear evolution equations

The holomorphic functional calculus for sectorial unbounded operators is an extension of the classical Dunford calculus for bounded operators. The interest in this calculus is motivated by the Kato square root problem and applications to the operator-sum method introduced by DaPrato and Grisvard to treat evolution equations on a finite interval. In this thesis we develop the holomorphic functional calculus for multisectorial and asymptotically bisectorial operators. We obtain versions of closed-sum theorems that allow to deduce maximal regularity for first and second order Cauchy problems both on the line and for the periodic problem. The results are then applied to prove existence and uniqueness of non-linear evolution equations.