Slice hyperholomorphic functional calculi

This paper is a survey on some functional calculi constructed with slice hyperholomorphic functions, mainly on the S-functional calculus. This is a functional calculus, defined for n-tuples of not necessarily commuting operators, and it is based on the recent theory of slice hyperholomorphic functions. Its version for commuting operators, called the SC-functional calculus, and its quaternionic version, called the quaternionic functional calculus, are presented, as well as the so-called F-functional calculus, based on the Fueter-Sce mapping theorem in integral form. Since the theory of hyperholomorphic functions is quite recent, the paper contains the main results of this function theory that are necessary to introduce these calculi, as well as the Fueter-Sce mapping theorem in integral form.