The Fueter mapping theorem in integral form and the ℱ‐functional calculus

In this paper we show a version of the Fueter mapping theorem that can be stated in integral form based on the Cauchy formulas for slice monogenic (or slice regular) functions. More precisely, given a holomorphic function f of a paravector variable, we generate a monogenic function by an integral transform whose kernel is particularly simple. This procedure allows us to define a functional calculus for n ‐tuples of commuting operators (called ℱ‐functional calculus) based on a new notion of spectrum, called ℱ‐spectrum, for the n ‐tuples of operators. Analogous results are shown for the quaternionic version of the theory and for the related ℱ‐functional calculus. Copyright © 2010 John Wiley & Sons, Ltd.