The $$\mathcal {F}$$-Resolvent Equation and Riesz Projectors for the $$\mathcal {F}$$-Functional Calculus

Abstract

AbstractThe Fueter-Sce-Qian mapping theorem gives a constructive way to extend holomorphic functions of one complex variable to slice hyperholomorphic functions. By means of the Cauchy formula for slice hyperholomorphic functions it is possible to have a Fueter-Sce-Qian mapping theorem in integral form for n odd. On this theorem it is based the $$ \mathcal {F}$$ F -functional calculus for n-tuples of commuting operators. It is a functional calculus based on the commutative version of the S spectrum. Furthermore, it is a monogenic functional calculus in the spirit of McIntosh and collaborators. In this paper, inspired by the quaternionic case and some particular Clifford algebras cases, we show a general resolvent equation for the $$ \mathcal {F}$$ F -functional calculus in the Clifford algebra setting. Moreover, we prove that the $$ \mathcal {F}$$ F -resolvent equation is the suitable equation to study the Riesz projectors.