# Spectral representations of normal operators via Intertwining Quaternionic Projection Valued Measures

February 08, 2016

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.

Keywords:

quaternionic functional analysis, spectral theory in quaternionic Hilbert spaces