The spectral theorem for quaternionic unbounded normal operators based on the S-spectrum

TL;DR

This paper establishes the spectral theorem for quaternionic unbounded normal operators using the S-spectrum, completing the spectral analysis foundation for quaternionic operators and impacting quaternionic quantum mechanics.

Contribution

It introduces a method to extend the spectral theorem from bounded to unbounded quaternionic normal operators via a transformation, based on the S-spectrum.

Findings

Proves the spectral theorem for quaternionic unbounded normal operators.

Shows the S-spectrum is the correct spectral object for quaternionic operators.

Provides a foundation for quaternionic spectral analysis and quantum mechanics applications.

Abstract

In this paper we prove the spectral theorem for quaternionic unbounded normal operators using the notion of $S$-spectrum. The proof technique consists of first establishing a spectral theorem for quaternionic bounded normal operators and then using a transformation which maps a quaternionic unbounded normal operator to a quaternionic bounded normal operator. With this paper we complete the foundation of spectral analysis of quaternionic operators. The $S$-spectrum has been introduced to define the quaternionic functional calculus but it turns out to be the correct object also for the spectral theorem for quaternionic normal operators. The fact that the correct notion of spectrum for quaternionic operators was not previously known has been one of the main obstructions to fully understanding the spectral theorem in this setting. A prime motivation for studying the spectral theorem for quaternionic unbounded normal operators is given by the subclass of unbounded anti-self adjoint quaternionic operators which play a crucial role in the quaternionic quantum mechanics.