Spectrum and Analytic Functional Calculus in Real and Quaternionic Frameworks

TL;DR

This paper develops a spectrum and analytic functional calculus for quaternionic linear operators, extending real operator results, using classical kernels and spectra in the complex plane, and introduces a joint spectrum for operator pairs.

Contribution

It introduces a refined analytic functional calculus for quaternionic operators based on classical methods and explores a joint spectrum for pairs of operators in this framework.

Findings

Constructed an analytic functional calculus for quaternionic operators.

Extended real linear operator spectrum results to quaternionic case.

Discussed a quaternionic joint spectrum for operator pairs.

Abstract

We present an approach to the spectrum and analytic functional calculus for quaternionic linear operators, following the corresponding results concerning the real linear operators. In fact, the construction of the analytic functional calculus for real linear operators can be refined to get a similar construction for quaternionic linear ones, in a classical manner, using a Riesz-Dunford-Gelfand type kernel, and considering spectra in the complex plane. A quaternionic joint spectrum for pairs of operators is also discussed, and an analytic functional calculus is constructed, via a Martinelli type kernel in two variables.