Formulations of the -functional calculus and some consequences

Abstract

In this paper we introduce the two possible formulations of the -functional calculus that are based on the Fueter–Sce mapping theorem in integral form and we introduce the pseudo--resolvent equation. In the case of dimension 3 we prove the -resolvent equation and we study the analogue of the Riesz projectors associated with this calculus. The case of dimension 3 is also useful to study the quaternionic version of the -functional calculus.