ABSTRACT
This chapter focuses on the study of semigroups of bounded operators introducing analytic semigroups and analyzes their main properties. It describes analytic semigroups and explains the definition of analytic semigroups via the Dunford integral, starting from the easiest class of semigroups (which are also analytic): the uniformly continuous semigroups. The chapter introduces the sectorial operators, which are the operators which are associated to analytic semigroups. It defines the analytic semigroup associated with the sectorial operator and also describes its main properties via the Dunford integral. The chapter shows in general analytic semigroups are not strongly continuous semigroups since the operator is not required to be densely defined in the Banach space. It provides equivalent characterization of the interpolation space, which plays a relevant role in the study of abstract Cauchy problems associated with sectorial operators. The chapter deals with a very useful (in view of applications) condition to guarantee that a closed operator is sectorial.
