On the Decomposition of Unitary Representations of Lie Groups
نویسنده
چکیده
a continuous1 unitary representation of G in a separable Hubert space ^ over the complex numbers. Let a be any element of the Lie algebra ® of G; then exp 6a, as defined in [l, chap. IV, §VIII] is an element of the group G and as 6 varies over the real line, exp 6a varies over a certain one-parameter subgroup of G. Therefore the operators Z7(exp 6a) form a one-parameter group of unitary operators in §. Hence there exists, as is well known, a self-adjoint (hypermaximal) operator A defined by lim^o (l/9)(U(exp 6a) — T)x = iAx where DA, the domain of definition of A, is the set of those x£^j for which this limit exists. And
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