A second look at "A Geometric Proof of the Spectral Theorem for Unbounded Self-Adjoint Operators"

TL;DR

This paper presents a new geometric proof of the spectral theorem for unbounded self-adjoint operators in Hilbert spaces, using a splitting into positive and negative parts to construct the spectral family.

Contribution

It introduces a novel geometric approach to the spectral theorem for unbounded self-adjoint operators, based on splitting the operator into positive and negative components.

Findings

Spectral family constructed from positive and negative parts

New geometric proof technique for spectral theorem

Applicable to unbounded self-adjoint operators

Abstract

A new geometric proof of the spectral theorem for unbounded self-adjoint operators A in a Hilbert space H is given based on a splitting of A in positive and negative parts A+ and A-. For both operators A+ and A- the spectral family can be defined immediately and then put together to become the spectral family of A.