A new approach to the S-functional calculus

TL;DR

This paper introduces a novel approach to the S-functional calculus for quaternionic operators, linking it to complex operator spectra and the Riesz-Dunford calculus, with simplified proofs of existing results.

Contribution

It establishes a connection between the S-spectrum of quaternionic operators and complex spectra, and shows the S-functional calculus can be derived from the Riesz-Dunford calculus.

Findings

S-spectrum is a union of spectra of complex operators

S-functional calculus is obtained via Riesz-Dunford calculus

Provides simplified proofs of known results

Abstract

In this paper, we first prove that the S-spectrum of a bounded right quaternionic linear operator on a two-sided quaternionic Banach space is a union of the spectrum of some bounded linear operators on a complex Banach space. Furthermore, we show that the S-functional calculus is obtained by the Riesz-Dunford functional calculus for complex linear operators. We also give simple proofs of some already existing results.