The $\mathcal{F}$-resolvent equation and Riesz projectors for the $\mathcal{F}$-functional calculus

TL;DR

This paper develops the $$-resolvent equation within Clifford algebras and demonstrates how the $$-functional calculus generates Riesz projectors, extending known quaternionic results to a more complex algebraic setting.

Contribution

It introduces a generalized $$-resolvent equation for Clifford algebras and proves that the $$-functional calculus produces Riesz projectors in this setting.

Findings

Generalization of the $$-resolvent equation to Clifford algebras.

Proof that the $$-functional calculus generates Riesz projectors.

Extension of quaternionic results to Clifford algebra context.

Abstract

The Fueter-Sce-Qian mapping theorem is a two steps procedure to extend holomorphic functions of one complex variable to quaternionic or Clifford algebra-valued functions in the kernel of a suitable generalized Cauchy-Riemann operator. Using the Cauchy formula of slice monogenic functions it is possible to give the Fueter-Sce-Qian extension theorem an integral form and to define the $\mathcal{F}$-functional calculus for $n$-tuples of commuting operators. This functional calculus is defined on the $S$-spectrum but it generates a monogenic functional calculus in the spirit of McIntosh and collaborators. One of the main goals of this paper is to show that the $\mathcal{F}$-functional calculus generates the Riesz projectors. The existence of such projectors is obtained via the $\mathcal{F}$-resolvent equation that we have generalized to the Clifford algebra setting. This equation was known in the quaternionic setting, but the Clifford algebras setting turned out to be much more complicated.