A polyanalytic functional calculus of order 2 on the 𝑆-spectrum

Martino, Antonino De; Pinton, Stefano

doi:10.1090/proc/16285

Cited by 4 publications

(13 citation statements)

References 32 publications

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“…The case of dimension five is different from what happens in the quaternionic case (see [12,29]), in which the Fueter map can be factorized only as DD and DD: here we obtain a reacher structure.…”

Section: Function Spaces Of Axial Type In Dimension Fivementioning

confidence: 73%

“…which is investigated in [29]. Even if the diagrams (4.1) and (4.2) come from the Fueter mapping theorem and the factorization of the Fueter operator T F 2 = ∆, we get two different fine structures.…”

Section: The Fine Structures In Dimension Fivementioning

confidence: 99%

“…In [13] the Q-functional calculus is introduced to get a product rule for the F -functional calculus. Moreover, we study a resolvent equation for this functional calculus suitable to generate the Riesz projectors in this setting and to get the following product rule, see [35].…”

Section: Harmonic Functional Calculus For Unbounded Operatorsmentioning

confidence: 99%

“…where AP 2 (Ω D ) is a space of polyanalytic functions. This structure is investigated in [29] together with its functional calculus.…”

Section: Introductionmentioning

confidence: 99%

“…In recent times it has been developed a new branch of the spectral theory on the S-spectrum that is called fine structures on the S-spectrum. It consists of the function spaces arising form the Fueter-Sce theorem with theirs integral representations that are used to define various functional calculi, see [13,16,34,35].…”

Section: Introductionmentioning

confidence: 99%

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Axially Harmonic Functions and the Harmonic Functional Calculus on the S-spectrum

et al. 2022

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The spectral theory on the S-spectrum was introduced to give an appropriate mathematical setting to quaternionic quantum mechanics, but it was soon realized that there were different applications of this theory, for example, to fractional heat diffusion and to the spectral theory for the Dirac operator on manifolds. In this seminal paper we introduce the harmonic functional calculus based on the S-spectrum and on an integral representation of axially harmonic functions. This calculus can be seen as a bridge between harmonic analysis and the spectral theory. The resolvent operator of the harmonic functional calculus is the commutative version of the pseudo S-resolvent operator. This new calculus also appears, in a natural way, in the product rule for the F-functional calculus.

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Section: Function Spaces Of Axial Type In Dimension Fivementioning

confidence: 73%

Section: The Fine Structures In Dimension Fivementioning

confidence: 99%

Section: Harmonic Functional Calculus For Unbounded Operatorsmentioning

confidence: 99%

“…where AP 2 (Ω D ) is a space of polyanalytic functions. This structure is investigated in [29] together with its functional calculus.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Axially Harmonic Functions and the Harmonic Functional Calculus on the S-spectrum

et al. 2022

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Properties of a polyanalytic functional calculus on the S‐spectrum

De Martino,

Pinton

2023

Mathematische Nachrichten

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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 Laplace operator in four real variables, namely, Δ, to a slice hyperholomorphic function. The polyanalytic functional calculus, that we study in this paper, is based on the factorization of . Instead of applying directly the Laplace operator to a slice hyperholomorphic function, we apply first the operator and we get a polyanalytic function of order 2, that is, a function that belongs to the kernel of . We can represent this type of functions in an integral form and then we can define the polyanalytic functional calculus on S‐spectrum. The main goal of this paper is to show the principal properties of this functional calculus. In particular, we study a resolvent equation suitable for proving a product rule and generate the Riesz projectors.

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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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A polyanalytic functional calculus of order 2 on the 𝑆-spectrum

Cited by 4 publications

References 32 publications

Axially Harmonic Functions and the Harmonic Functional Calculus on the S-spectrum

Axially Harmonic Functions and the Harmonic Functional Calculus on the S-spectrum

Properties of a polyanalytic functional calculus on the S‐spectrum

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

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