A note on semi-groups of unbounded self-adjoint operators

In this note we extend a theorem about integral representations of semi-groups of bounded self-adjoint operators to a theorem about semi-groups of unbounded self-adjoint operators. The theorem for semi-groups of bounded self-adjoint operators has been proved in different ways by B. v. Sz. Nagy [4; 5], E. Hille [3 ], and also follows from a more general theorem of S. Bochner [1]. Semi-groups of unbounded self-adjoint operators arose in quite a natural way in an investigation of the author [2] on positive definite functions. In the following theorem DA shall represent the domain of an operator A which is defined in a Hilbert space.