ABSTRACT
This chapter introduces the concept of semigroup of bounded linear operators and explains two important classes of semigroups: the strongly continuous and the analytic semigroups. Semigroups of bounded operators naturally arise in the study of parabolic equations: a suitable semigroup governs the dynamics of parabolic equations. The chapter shows that semigroups are a very powerful tool in the study of linear parabolic equations. It describes the strongly continuous semigroups and proves some basic properties. The chapter details the concept of the infinitesimal generator of a strongly continuous semigroup and states its main properties: this operator is densely defined and its resolvent set contains a right-halfplane. It focuses on the well celebrated Hille-Yosida, Lumer-Phillips and Trotter-Kato theorems. The chapter also explains abstract Cauchy problems associated to generators of strongly continuous semigroups. It contains some exercises which may help the reader to become more familiar with strongly continuous semigroups.
