Semigroup Methods in Mathematical Physics

Abstract

The theory of one-parameter semigroups of operators in Banach spaces is concerned with the study of the operator-valued exponential function, its representation and properties, in infinite-dimensional function spaces. The theory can also be considered as a generalization of Stone’s theorem33 on the representation of one-parameter groups of operators in Hilbert space. The theory of semigroups of operators had its origin in 1936, when Hille10 investigated certain concrete semigroups of operators. In 1948, Hille11 published his treatise on semigroups of operators; and in the 1950’s this treatise stimulated a considerable amount of research on semigroups and their applications. In 1957, Hille and Phillips12 published an extensive revision of Hille’s treatise, and at the present time their treatise is the bible of workers in semigroup theory. In addition to the Hille—Phillips treatise, semigroups of operators are the subject of lecture notes by the author2 and Yosida40; chapters on semigroups of operators can be found in the books of Dunford and Schwartz,5 Maurin,20 and Riesz and Sz.Nagy.30 We also refer to the papers of Phillips,27,29 the first of which presents a survey of semigroup theory, while the second gives a very readable introduction to semigroup theory and its applications in the theory of partial differential equations.