Slice monogenic functions of a Clifford variable via the $S$-functional calculus

In some recent papers (called $\mathcal{S}$-functional calculus) for n-tuples of both bounded and unbounded not-necessarily commuting operators. The $\mathcal{S}$-functional calculus is based on the notion of $\mathcal{S}$-spectrum, which naturally arises from the definition of the $\mathcal{S}$-resolvent operator for n-tuples of operators. The $\mathcal{S}$-resolvent operator plays the same role as the usual resolvent operator for the Riesz–Dunford functional calculus, which is associated to a complex operator acting on a Banach space. When one considers commuting operators (bounded or unbounded) there is the possibility of simplifying the computation of the $\mathcal{S}$-spectrum. In fact, in this case we can use the F-spectrum, which is easier to compute than the $\mathcal{S}$-spectrum. In the case of commuting operators, our functional calculus is based on the $\mathcal{F}$-spectrum and will be called $\mathcal{SC}$-functional calculus. We point out that for a correct definition of the $\mathcal{S}$-resolvent operator and of the $\mathcal{SC}$-resolvent operator in the unbounded case we have to face different extension problems. Another reason for a more detailed study of the $\mathcal{F}$-spectrum is that it is related to the $\mathcal{F}$-functional calculus which is based on the integral version of the Fueter mapping theorem. This functional calculus is associated to monogenic functions constructed by starting from slice monogenic functions.

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Slice monogenic functions of a Clifford variable via the 𝑆-functional calculus

Proceedings of the American Mathematical Society Series B ◽

10.1090/bproc/94 ◽

2021 ◽

Vol 8 (23) ◽

pp. 281-296

Author(s):

Fabrizio Colombo ◽

David Kimsey ◽

Stefano Pinton ◽

Irene Sabadini

Keyword(s):

Functional Calculus ◽

Clifford Algebras ◽

Function Theory ◽

Monogenic Functions ◽

Content Type ◽

New Class ◽

Zero Divisors ◽

Slice Monogenic Functions ◽

Class Of Functions ◽

Clifford Numbers

In this paper we define a new function theory of slice monogenic functions of a Clifford variable using the S S -functional calculus for Clifford numbers. Previous attempts of such a function theory were obstructed by the fact that Clifford algebras, of sufficiently high order, have zero divisors. The fact that Clifford algebras have zero divisors does not pose any difficulty whatsoever with respect to our approach. The new class of functions introduced in this paper will be called the class of slice monogenic Clifford functions to stress the fact that they are defined on open sets of the Clifford algebra R n \mathbb {R}_n . The methodology can be generalized, for example, to handle the case of noncommuting matrix variables.

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Distributions and the Global Operator of Slice Monogenic Functions

Complex Analysis and Operator Theory ◽

10.1007/s11785-013-0322-6 ◽

2013 ◽

Vol 8 (6) ◽

pp. 1257-1268 ◽

Cited By ~ 5

Author(s):

Fabrizio Colombo ◽

Franciscus Sommen

Keyword(s):

Monogenic Functions ◽

Global Operator ◽

Slice Monogenic Functions

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A nonconstant coefficients differential operator associated to slice monogenic functions

Transactions of the American Mathematical Society ◽

10.1090/s0002-9947-2012-05689-3 ◽

2012 ◽

Vol 365 (1) ◽

pp. 303-318 ◽

Cited By ~ 25

Author(s):

Fabrizio Colombo ◽

J. Oscar González-Cervantes ◽

Irene Sabadini

Keyword(s):

Differential Operator ◽

Monogenic Functions ◽

Slice Monogenic Functions

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The Bergman-Sce transform for slice monogenic functions

Mathematical Methods in the Applied Sciences ◽

10.1002/mma.1489 ◽

2011 ◽

Vol 34 (15) ◽

pp. 1896-1909 ◽

Cited By ~ 8

Author(s):

Fabrizio Colombo ◽

Jose O. Gonzàles Cervantes ◽

Irene Sabadini

Keyword(s):

Monogenic Functions ◽

Slice Monogenic Functions

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The Radon transform between monogenic and generalized slice monogenic functions

Mathematische Annalen ◽

10.1007/s00208-015-1182-3 ◽

2015 ◽

Vol 363 (3-4) ◽

pp. 733-752 ◽

Cited By ~ 12

Author(s):

F. Colombo ◽

R. Lávička ◽

I. Sabadini ◽

V. Souček

Keyword(s):

Radon Transform ◽

Monogenic Functions ◽

Slice Monogenic Functions

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An Introduction to Hyperholomorphic Spectral Theories and Fractional Powers of Vector Operators

Advances in Applied Clifford Algebras ◽

10.1007/s00006-021-01148-1 ◽

2021 ◽

Vol 31 (3) ◽

Author(s):

Fabrizio Colombo ◽

Jonathan Gantner ◽

Stefano Pinton

Keyword(s):

Spectral Theory ◽

Functional Calculus ◽

Natural Extension ◽

Several Complex Variables ◽

Monogenic Functions ◽

Fractional Powers ◽

Holomorphic Functional Calculus ◽

Slice Hyperholomorphic Functions ◽

Hyperholomorphic Functions ◽

Nonhomogeneous Materials

AbstractThe 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 1 , … , 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.

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