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

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