Semi-Groups of Contraction Operators

Abstract

The theory of semi-groups of linear operators had its origin in Stone's theorem on groups of unitary operators acting in a Hilbert space (1932). Stone's theorem was motivated by the time dependent solution of Schroedinger's wave equation for quantum mechanics. The study of semi-groups of operators, rather than groups of operators, was undertaken by Hillein 1936 who became interested in the semi-group properties of certain classical singular integrals. However it was not until 1948 that the applicability of the theory was fully appreciated. At that time K. Yosida applied semigroup methods to the diffusion equation. In the hands of Feller, Hille, Kendall, Reuter, and Yosida, the theory became an integral part of the theory of probability. It has also been applied with profit to the Cauchy problem for the wave equation; here the contributions of Friedrichs, Lax, and Phillips should be mentioned.

We shall use the initial value problems of mathematical physics to motivate the theory of semi-groups. A suitably abstract formulation canbe obtained by the following considerations.