One-Parameter Groups and Semigroups of Operators

Abstract

Chapter 6 gives a concise introduction into the theory of one-parameter groups or semigroups of operators with an emphasis on the interplay between groups and semigroups and their generators. In the first section, one-parameter unitary groups are investigated, and two fundamental theorems, Stone’s theorem and Trotter’s formula, are proved. Semigroups of operators are applied to Cauchy problems for abstract differential equations on Hilbert space. Then generators of semigroups of contractions on Banach spaces are studied, and the Hille–Yosida theorem is proved. Finally, generators of contraction semigroups on Hilbert space are characterized as m-dissipative operators.