A Panorama on Quaternionic Spectral Theory and Related Functional Calculi

In this paper we offer an overview of the state of the art of quaternionic spectral theory. Precisely we review some functional calculi and the quaternionic spectral theorem based on the S-spectrum. We start with the S-functional calculus which is the Riesz-Dunford functional calculus for quaternionic operators which suggested the existence of the S-spectrum of quaternionic operators, then we introduce the Spectral Theorem based on the S-spectrum which is of fundamental importance for the formulation of quaternionic quantum mechanics. Moreover we discuss the quaternionic H-infinity-functional calculus that is the quaternionic analogue of the H-infinity-functional calculus for sectorial operators introduced by A. McIntosh. In the case a quaternionic linear operator is the infinitesimal generator quaternionic group of linear operators by the Laplace-Stieltjes transform we extend the Philips functional calculus in this setting. The W-functional calculus and the F-functional calculus are monogenic functional calculi, in the spirit of the monogenic functional calculus introduced by A. McIntosh, but both calculi are based on slice hyperholomorphic functions and on manipulations of their Cauchy formulas.