Abstract
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.
Mathematics Subject Classification (2000). Primary 47A10, 47A60.
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Colombo, F., Sabadini, I. (2012). An Invitation to the S-functional Calculus. In: Arendt, W., Ball, J., Behrndt, J., Förster, KH., Mehrmann, V., Trunk, C. (eds) Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations. Operator Theory: Advances and Applications, vol 221. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0297-0_13
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