Abstract
The aim of this paper is to give an overview of the spectral theories associated with the notions of holomorphicity in dimension greater than one. A first natural extension is the theory of several complex variables whose Cauchy formula is used to define the holomorphic functional calculus for n-tuples of operators (A_1,ldots ,A_n). A second way is to consider hyperholomorphic functions of quaternionic or paravector variables. In this case, by the Fueter-Sce-Qian mapping theorem, we have two different notions of hyperholomorphic functions that are called slice hyperholomorphic functions and monogenic functions. Slice hyperholomorphic functions generate the spectral theory based on the S-spectrum while monogenic functions induce the spectral theory based on the monogenic spectrum. There is also an interesting relation between the two hyperholomorphic spectral theories via the F-functional calculus. The two hyperholomorphic spectral theories have different and complementary applications. We finally discuss how to define the fractional Fourier’s law for nonhomogeneous materials using the spectral theory on the S-spectrum.
Highlights
The problem to define functions of an operator A or of an n-tuple of operators (A1, . . . , An) is very important both in mathematics and in physics and has been investigated with different methods starting from the beginning of theThis article is part of the Topical Collection on Proceedings ICCA 12, Hefei, 2020, edited by Guangbin Ren, Uwe Kahler, Rafal Ablamowicz, Fabrizio Colombo, Pierre Dechant, Jacques Helmstetter, G
The Cauchy formula allows to define the holomorphic functional calculus in Banach spaces [14], and this calculus can be extended to unbounded operators
We show how the Fueter-Sce mapping theorem provides an alternative way to define the functional calculus for monogenic functions
Summary
The problem to define functions of an operator A or of an n-tuple of operators (A1, . . . , An) is very important both in mathematics and in physics and has been investigated with different methods starting from the beginning of theThis article is part of the Topical Collection on Proceedings ICCA 12, Hefei, 2020, edited by Guangbin Ren, Uwe Kahler, Rafal Ablamowicz, Fabrizio Colombo, Pierre Dechant, Jacques Helmstetter, G. An) is very important both in mathematics and in physics and has been investigated with different methods starting from the beginning of the. The spectral theorem is one of the most important tools to define functions of normal operators on a Hilbert space and it is of crucial importance in quantum mechanics as well as the Weyl functional calculus. The theory of holomorphic functions plays a central role in operator theory. The Cauchy formula allows to define the holomorphic functional calculus (often called Riesz-Dunford functional calculus) in Banach spaces [14], and this calculus can be extended to unbounded operators. For sectorial operators the H∞-functional calculus, introduced by A. McIntosh in [71], turned out to be the most important extension
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