16 - Semigroups of Operators

Summary

The chapter is a gentle introduction to the theory of strongly continuous semigroups of operators. We present the notion of the generator, discuss the generator’s basic properties and study a number of examples. We learn that the way to discover whether a given operator is a semigroup generator is by examining the resolvent equation, and are thus naturally led to the Hille–Yosida–Feller–Phillips–Miyadera theorem that characterizes generators in terms of resolvents. Two valuable consequences, the generation theorems for maximal dissipative operators in Hilbert space and operators satisfying the positive-maximum principle in the space of continuous function, are also explained. This material is supplemented with three theorems on the generation of positive semigroups. The reader of this book, however, will undoubtedly have noticed that the whole theory would have failed were it not for the fact that we are working in Banach spaces; without the assumption of completeness, we could not be sure that the Yosida approximation converges, and the entire reasoning would have collapsed.