Spectral representations of normal operators via Intertwining Quaternionic Projection Valued Measures

TL;DR

This paper explores spectral representations of quaternionic normal operators using intertwining quaternionic projection-valued measures, advancing quaternionic quantum mechanics and addressing tensor product issues.

Contribution

It introduces the concept of intertwining quaternionic PVMs and links them to spectral features of quaternionic normal operators, providing new spectral notions.

Findings

Existence of spectral notions equivalent to spherical spectrum

Development of intertwining quaternionic PVMs

Connection between PVMs and spectral properties of operators

Abstract

The possibility of formulating quantum mechanics over quaternionic Hilbert spaces can be traced back to von Neumann's foundational works in the thirties. The absence of a suitable quaternionic version of spectrum prevented the full development of the theory. The first rigorous quaternionic formulation has been started only in 2007 with the definition of the spherical spectrum of a quaternionic operator based on a quadratic version of resolvent operator. The relevance of this notion is proved by the existence of a quaternionic continuous functional calculus and a theory of quaternionic semigroups relying upon it. A problem of quaternionic formulation is the description of composite quantum systems in absence of a natural tensor product due to non-commutativity of quaternions. A promising tool towards a solution is a quaternionic projection-valued measure (PVM), making possible a tensor product of quaternionic operators with physical relevance. A notion with this property, called intertwining quaternionic PVM, is presented here. This foundational paper aims to investigate the interplay of this new mathematical object and the spherical spectral features of quaternionic generally unbounded normal operators. We discover in particular the existence of other spectral notions equivalent to the spherical ones, but based on a standard non-quadratic notion of resolvent operator.