Properties of a polyanalytic functional calculus on the <i>S</i>‐spectrum

De Martino, Antonino; Pinton, Stefano

doi:10.1002/mana.202200318

Mathematische Nachrichten

2023

DOI: 10.1002/mana.202200318

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Properties of a polyanalytic functional calculus on the S‐spectrum

Antonino De Martino,

Stefano Pinton

Abstract: The Fueter mapping theorem gives a constructive way to extend holomorphic functions of one complex variable to monogenic functions, that is, null solutions of the generalized Cauchy–Riemann operator in , denoted by . This theorem is divided in two steps. In the first step, a holomorphic function is extended to a slice hyperholomorphic function. The Cauchy formula for this type of functions is the starting point of the S‐functional calculus. In the second step, a monogenic function is obtained by applying the L… Show more

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Cited by 3 publications

References 30 publications

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The $$H^\infty $$-Functional Calculi for the Quaternionic Fine Structures of Dirac Type

Colombo,

Pinton,

Schlosser

2024

Milan J. Math.

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In recent works, various integral representations have been proposed for specific sets of functions. These representations are derived from the Fueter–Sce extension theorem, considering all possible factorizations of the Laplace operator in relation to both the Cauchy–Fueter operator (often referred to as the Dirac operator) and its conjugate. The collection of these function spaces, along with their corresponding functional calculi, are called the quaternionic fine structures within the context of the S-spectrum. In this paper, we utilize these integral representations of functions to introduce novel functional calculi tailored for quaternionic operators of sectorial type. Specifically, by leveraging the aforementioned factorization of the Laplace operator, we identify four distinct classes of functions: slice hyperholomorphic functions (leading to the S-functional calculus), axially harmonic functions (leading to the Q-functional calculus), axially polyanalytic functions of order 2 (leading to the $$P_2$$ P 2 -functional calculus), and axially monogenic functions (leading to the F-functional calculus). By applying the respective product rule, we establish the four different $$H^\infty $$ H ∞ -versions of these functional calculi.

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The $$H^\infty $$-Functional Calculi for the Quaternionic Fine Structures of Dirac Type

Colombo,

Pinton,

Schlosser

2024

Milan J. Math.

Self Cite

View full text Add to dashboard Cite

show abstract

The Fine Structure of the Spectral Theory on the S-Spectrum in Dimension Five

et al. 2023

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Holomorphic functions play a crucial role in operator theory and the Cauchy formula is a very important tool to define the functions of operators. The Fueter–Sce–Qian extension theorem is a two-step 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, contains the Weyl functional calculus as a particular case. In this paper we show that the extension operator from slice hyperholomorphic functions to monogenic functions admits various possible factorizations that induce different function spaces. The integral representations in such spaces allow to define the associated functional calculi based on the S-spectrum. The function spaces and the associated functional calculi define the so-called fine structure of the spectral theories on the S-spectrum. Among the possible fine structures there are the harmonic and polyharmonic functions and the associated harmonic and polyharmonic functional calculi. The study of the fine structures depends on the dimension considered and in this paper we study in detail the case of dimension five, and we describe all of them. The five-dimensional case is of crucial importance because it allows to determine almost all the function spaces will also appear in dimension greater than five, but with different orders.

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Harmonic and polyanalytic functional calculi on the S-spectrum for unbounded operators

Colombo,

De Martino,

Pinton

2023

Banach J. Math. Anal.

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Harmonic and polyanalytic functional calculi have been recently defined for bounded commuting operators. Their definitions are based on the Cauchy formula of slice hyperholomorphic functions and on the factorization of the Laplace operator in terms of the Cauchy–Fueter operator $${\mathcal{D}}$$ D and of its conjugate $$\overline{{\mathcal{D}}}.$$ D ¯ . Thanks to the Fueter extension theorem, when we apply the operator $${\mathcal{D}}$$ D to slice hyperholomorphic functions, we obtain harmonic functions and via the Cauchy formula of slice hyperholomorphic functions, we establish an integral representation for harmonic functions. This integral formula is used to define the harmonic functional calculus on the S-spectrum. Another possibility is to apply the conjugate of the Cauchy–Fueter operator to slice hyperholomorphic functions. In this case, with a similar procedure we obtain the class of polyanalytic functions, their integral representation, and the associated polyanalytic functional calculus. The aim of this paper is to extend the harmonic and the polyanalytic functional calculi to the case of unbounded operators and to prove some of the most important properties. These two functional calculi belong to so called fine structures on the S-spectrum in the quaternionic setting. Fine structures on the S-spectrum associated with Clifford algebras constitute a new research area that deeply connects different research fields such as operator theory, harmonic analysis, and hypercomplex analysis.

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Properties of a polyanalytic functional calculus on the S‐spectrum

Cited by 3 publications

References 30 publications

The $$H^\infty $$-Functional Calculi for the Quaternionic Fine Structures of Dirac Type

The $$H^\infty $$-Functional Calculi for the Quaternionic Fine Structures of Dirac Type

The Fine Structure of the Spectral Theory on the S-Spectrum in Dimension Five

Harmonic and polyanalytic functional calculi on the S-spectrum for unbounded operators

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