A nonstandard approach to the direct integral version of the Spectral Theorem
Isaac Goldbring, Fabrice Nonez
TL;DR
The paper develops a nonstandard-analysis framework to prove the direct integral version of the Spectral Theorem for unbounded self-adjoint operators. By constructing a hyperfinite internal model $(\tilde{H},\tilde{A})$ and passing to standard parts via Loeb measures, it obtains a direct integral decomposition $H \cong \int^{\oplus}_{\mathbb{R}} H_t \,d\mu(t)$ with an intertwining isometry $U$ such that $U(Ax)(t)=t\,U(x)(t)$ for $\mu$-a.e. $t$. The approach handles both real and complex Hilbert spaces without Cayley transforms and yields a nonstandard proof of the spectral-measure version. A key outcome is that $A$ is unitarily equivalent to the multiplication operator on the direct integral when $A$ is self-adjoint, and the spectral-measure representation $A=\int_{\mathbb{R}} \operatorname{id}_{\mathbb{R}} dP$ follows from a projected measure $P$ built from internal data. The work highlights how hyperfinite samplings, Loeb measure techniques, and internal direct integrals can illuminate spectral theory and offer a robust, uniform framework beyond the complex-field Casimir approach.
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.