A nonstandard approach to the direct integral version of the Spectral Theorem

Abstract

We use nonstandard methods to prove the direct integral version of the Spectral Theorem for Unbounded Self-adjoint Operators. Our proof avoids the standard reduction to the case of bounded normal operators via the Cayley transform and, as such, works uniformly for both real and complex Hilbert spaces. Our method also yields a new nonstandard proof of the spectral measure version of the Spectral Theorem.