One-Parameter Semigroups of Isometric Operators in Hilbert Space

Abstract

The theory of one-parameter groups of unitary operators in Hilbert Space due to Stone' shows that to such a group there corresponds a selfadjoint operator which is the infinitesimal operator of the group, and a resolution of the identity which is the resolution of the identity of that operator. The theory of sets of isometric operators with the group multiplicative properties has not been worked out; the object of this paper is to develop this theory. The basis for it is provided by my paper on Symmetrical Operators in Hilbert Space2 in which it is shown that a maximal symmetric operator is the infinitesimal generator of a semigroup of isometric operators. This paper handles the converse problem of showing that a semigroup with suitable multiplicative properties leads to a maximal symmetric operator as infinitesimal generator. The main results of the paper are as follows. We deal with a set of isometric operators U(t), defined for every t 2 0, and with all Hilbert Space e as domain for all t > 0. A. For all t > 0, s >a 0