Non commutative functional calculus: unbounded operators

TL;DR

This paper extends a quaternionic functional calculus from bounded to unbounded operators, addressing key differences and introducing a new eigenvalue equation for operators with non-real spectra.

Contribution

It develops a functional calculus for unbounded quaternionic operators, expanding previous bounded operator results and introducing a new eigenvalue equation for non-real spectra.

Findings

Extended functional calculus to unbounded operators

Identified key differences between bounded and unbounded cases

Proposed a new eigenvalue equation for non-real spectra

Abstract

In a recent work, \cite{cgss}, we developed a functional calculus for bounded operators defined on quaternionic Banach spaces. In this paper we show how the results from \cite{cgss} can be extended to the unbounded case, and we highlight the crucial differences between the two cases. In particular, we deduce a new eigenvalue equation, suitable for the construction of a functional calculus for operators whose spectrum is not necessarily real.