An Invitation to the S-functional Calculus
In this paper we give an overview of the S-functional calculus which is based on the Cauchy formula for slice monogenic functions.S uch a functional calculus works for n-tuples of noncommuting operators and it is based on the notion of S-spectrum.Th ere is a commutative version of the S-functional calculus, due to the fact that the Cauchy formula for slice monogenic functions admits two representations of the Cauchy kernel.W e will call SC-functional calculus the commutative version of the S-functional calculus. This version has the advantage that it is based on the notion of ℱ-spectrum, which turns out to be more simple to compute with respect to the S-spectrum. For commuting operators the two spectra are equal, but when the operators do not commute among themselves the ℱ-spectrum is not well defined.W e finally briefly introduce the main ideas on which the ℱ-functional calculus is inspired.T his functional calculus is based on the integral version of the Fueter-Sce mapping theorem and on the ℱ-spectrum.
KeywordsFunctional calculus for n-tuples of commuting operators the S-spectrum the ℱ-spectrum S-functional calculus S-functional calculus SC-functional calculus ℱ-functional spectrum
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