Introduction

Since the discovery of the S-spectrum in 2006 the theory of slice hyperholomorphic functions has become the underlining function theory on which new functional calculi for quaternionic operators and for n-tuples of operators have been developed. These calculi are the S-functional calculus and the F-functional calculus, and both are based on the notion of S-spectrum. In 2014 it was proved that the quaternionic spectral theorem is also naturally based on the S-spectrum. These facts restore the analogy with the classical case in which the holomorphic functional calculus and the spectral theorem are based on the same notion of spectrum.