Spectral theories on the S-Spectrum

One of the deepest results in hypercomplex analysis is the Fueter extension theorem. It gives a two steps procedure to extend holomorphic functions to the hyperholomorphic setting. The first step gives the class of slice hyperholomorphic functions; their Cauchy formula allows to define the so-called S-functional calculus for noncommuting operators based on the S-spectrum. In the second step, this extension procedure generates monogenic functions; the related monogenic functional calculus, based on the monogenic spectrum, was widely studied by McIntosh and collaborators.
In this talk, I will discuss the main notions of the S-spectrum and some concepts of the monogenic functional calculus. Moreover, I will also give some ideas on the new research direction of the fine structures.