On the decomposition of unitary representations of Lie groups

Abstract

a continuous' unitary representation of G in a separable Hilbert space & over the complex numbers. Let a be any element of the Lie algebra 65 of G; then exp Oa, as defined in [1, chap. IV, ?VIII] is an element of the group G and as 0 varies over the real line, exp Oa varies over a certain one-parameter subgroup of G. Therefore the operators U(exp Oa) form a one-parameter group of unitary operators in S. Hence there exists, as is well known, a self-adjoint (hypermaximal) operator A defined by limo,o (1/0) ( U(exp Oa) I)x = iAx where DA, the domain of definition of A, is the set of those xFt for which this limit exists. And