Normal operators in real and quaternionic Hilbert spaces

TL;DR

This paper introduces a new approach to understanding normal operators in real and quaternionic Hilbert spaces, deriving spectral representations and analyzing spectra using classical complex analysis methods.

Contribution

It presents a novel approach to spectral representation of normal operators in real Hilbert spaces and extends these results to quaternionic normal operators.

Findings

Spectral representation derived from the complex case for real Hilbert spaces.

Quaternionic normal operators analyzed via their relation to real normal operators.

Spectrum and measures of quaternionic operators are described using classical complex analysis.

Abstract

A new approach to normal operators in real Hilbert spaces is discussed, and a spectral representation is obtained, derived directly from the complex case. The results are then applied to quaternionic normal operators, regarded as a special class of real normal operators. This point of view allows us to consider their spectrum and associated measures to be defined on subsets of the complex plane, in a classical manner.