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

TL;DR

This paper introduces a nonstandard approach to prove the direct integral form of the Spectral Theorem for unbounded self-adjoint operators, providing a uniform proof for real and complex Hilbert spaces and a new perspective on spectral measures.

Contribution

It presents a novel nonstandard proof of the Spectral Theorem's direct integral and spectral measure versions, avoiding traditional reduction methods.

Findings

Provides a uniform nonstandard proof for real and complex Hilbert spaces.

Avoids reduction to bounded operators via Cayley transform.

Offers new insights into spectral measures of unbounded operators.

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.