Abstract. A spectral theorem for a normal operator on a real Hilbert space is proved by using the techniques of Banach algebras. This gives a unified treatment for the theory of normal operators on real, complex and quaternionic Hilbert spaces. A well known spectral theorem for a normal operator on a complex Hilbert space is deduced. An example is given to illustrate a difference between the behaviour of operators on a real and a complex Hilbert space.

AMS Subject Classification (1991): 46L89, 47B15, 46L05

Received February 28, 1994 and in revised form May 17, 1994. (Registered under 5597/2009.)