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Functions and operators of the polyharmonic and polyanalytic Clifford fine structures on the $S$-spectrum

Fabrizio Colombo, Antonino De Martino, Stefano Pinton

TL;DR

This work develops a unified framework of Clifford fine structures on the $S$-spectrum, connecting slice hyperholomorphic, axially monogenic, polyharmonic, polyanalytic, and holomorphic Cliffordian function classes via integral representations and a family of compatible functional calculi. It introduces and interrelates the $S$-functional calculus, the $F$-functional calculus, and the monogenic calculus, proving their equivalence in appropriate settings and extending these ideas to polyharmonic and holomorphic Cliffordian contexts through Fueter–Sce mappings. The paper provides integral kernels and resolvent operators for axially polyharmonic and holomorphic Cliffordian functions, establishing well-defined, domain- and coordinate-independent calculi that generalize classical spectral theory to fully Clifford-algebra settings. These fine-structure calculi enable systematic operator-function constructions for Clifford-valued operators and have potential applications in quaternionic and higher-dimensional spectral problems, fractional diffusion, and quantum-mechanical operator theory. Overall, the results show that distinct analytic approaches to Clifford-analytic functional calculus converge to the same operator in the monogenic setting, and they extend to broader function spaces via Fueter–Sce factorization and integral representations on the $S$-spectrum.

Abstract

The spectral theory on the $S$-spectrum originated to give quaternionic quantum mechanics a precise mathematical foundation and as a spectral theory for linear operators in vector analysis. This theory has proven to be significantly more general than initially anticipated, naturally extending to fully Clifford operators and revealing unexpected connections with the spectral theory based on the monogenic spectrum, developed over forty years ago by A. McIntosh and collaborators. In recent years, we have combined slice hyperholomorphic functions with the Fueter-Sce mapping theorem, also called Fueter-Sce extension theorem, to broaden the class of functions and operators to which the theory can be applied. This generalization has led to the definition of what we call the {\em fine structures on the $S$-spectrum}, consisting of classes of functions that admit an integral representation and their associated functional calculi. In this paper, we focus on the fine structures within the Clifford algebra setting, particularly addressing polyharmonic functions, polyanalytic functions, holomorphic Cliffordian functions and their associated functional calculi defined via integral representation formulas. Moreover, we demonstrate that the monogenic functional calculus, defined via the monogenic Cauchy formula, and the $F$-functional calculus of the fine structures, defined via the Fueter-Sce mapping theorem in integral form, yield the same operator.

Functions and operators of the polyharmonic and polyanalytic Clifford fine structures on the $S$-spectrum

TL;DR

This work develops a unified framework of Clifford fine structures on the -spectrum, connecting slice hyperholomorphic, axially monogenic, polyharmonic, polyanalytic, and holomorphic Cliffordian function classes via integral representations and a family of compatible functional calculi. It introduces and interrelates the -functional calculus, the -functional calculus, and the monogenic calculus, proving their equivalence in appropriate settings and extending these ideas to polyharmonic and holomorphic Cliffordian contexts through Fueter–Sce mappings. The paper provides integral kernels and resolvent operators for axially polyharmonic and holomorphic Cliffordian functions, establishing well-defined, domain- and coordinate-independent calculi that generalize classical spectral theory to fully Clifford-algebra settings. These fine-structure calculi enable systematic operator-function constructions for Clifford-valued operators and have potential applications in quaternionic and higher-dimensional spectral problems, fractional diffusion, and quantum-mechanical operator theory. Overall, the results show that distinct analytic approaches to Clifford-analytic functional calculus converge to the same operator in the monogenic setting, and they extend to broader function spaces via Fueter–Sce factorization and integral representations on the -spectrum.

Abstract

The spectral theory on the -spectrum originated to give quaternionic quantum mechanics a precise mathematical foundation and as a spectral theory for linear operators in vector analysis. This theory has proven to be significantly more general than initially anticipated, naturally extending to fully Clifford operators and revealing unexpected connections with the spectral theory based on the monogenic spectrum, developed over forty years ago by A. McIntosh and collaborators. In recent years, we have combined slice hyperholomorphic functions with the Fueter-Sce mapping theorem, also called Fueter-Sce extension theorem, to broaden the class of functions and operators to which the theory can be applied. This generalization has led to the definition of what we call the {\em fine structures on the -spectrum}, consisting of classes of functions that admit an integral representation and their associated functional calculi. In this paper, we focus on the fine structures within the Clifford algebra setting, particularly addressing polyharmonic functions, polyanalytic functions, holomorphic Cliffordian functions and their associated functional calculi defined via integral representation formulas. Moreover, we demonstrate that the monogenic functional calculus, defined via the monogenic Cauchy formula, and the -functional calculus of the fine structures, defined via the Fueter-Sce mapping theorem in integral form, yield the same operator.
Paper Structure (17 sections, 101 theorems, 437 equations)

This paper contains 17 sections, 101 theorems, 437 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Slice monogenic functions
  4. The $S$-functional calculus for $n$-tuples of operators
  5. The monogenic functional calculi and their connections
  6. The F-functional calculus
  7. The monogenic functional calculus
  8. Equivalence between monogenic and the F-functional calculi
  9. Integral representation of axially polyharmonic functions
  10. A Polyharmonic functional calculus based on the $S$-spectrum
  11. Holomorphic Cliffordian functions in integral form
  12. The holomorphic Cliffordian functional calculus based on the $S$-spectrum
  13. Product rule for the $F$-functional calculus
  14. Integral representation of polyanalytic functions
  15. Polyanalytic functional calculus based on the $S$-spectrum
  16. ...and 2 more sections

Key Result

Theorem 2.8

Let $U \subset \mathbb{R}^{n+1}$ be an open set and $I \in \mathbb{S}$. We assume that $f$ is a left slice monogenic function and $g$ is a right slice monogenic function in $U$. Furthermore, let $D_I \subset U \cap \mathbb{C}_I$ be an open and bounded subset of $\mathbb{C}_I$ with $\overline{D}_I \s

Theorems & Definitions (217)

  • Definition 2.1
  • Definition 2.3
  • Definition 2.4: Slice Cauchy domain
  • Definition 2.5: Slice hyperholomorphic functions (or slice monogenic functions)
  • Definition 2.6
  • Theorem 2.8
  • Proposition 2.9: Cauchy kernel series
  • Proposition 2.10
  • Definition 2.11
  • Theorem 2.14: The Cauchy formulas for slice monogenic functions, see CGKColomboSabadiniStruppa2011
  • ...and 207 more