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The $H^\infty$-functional calculi for the quaternionic fine structures of Dirac type

Fabrizio Colombo, Stefano Pinton, Peter Schlosser

Abstract

In this paper, we utilize various integral representations derived from the Fueter-Sce extension theorem, to introduce novel functional calculi tailored for quaternionic operators of sectorial type. Specifically, due to the different factorizations of the Laplace opertor with respect to the Cauchy-Fueter operator and its conjugate, 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 $P_2$-functional calculus), and axially monogenic functions (leading to the $F$-functional calculus). By applying the respective product rule, we establish the four different $H^\infty$-versions of these functional calculi.

The $H^\infty$-functional calculi for the quaternionic fine structures of Dirac type

Abstract

In this paper, we utilize various integral representations derived from the Fueter-Sce extension theorem, to introduce novel functional calculi tailored for quaternionic operators of sectorial type. Specifically, due to the different factorizations of the Laplace opertor with respect to the Cauchy-Fueter operator and its conjugate, we identify four distinct classes of functions: Slice hyperholomorphic functions (leading to the -functional calculus), axially harmonic functions (leading to the -functional calculus), axially polyanalytic functions of order (leading to the -functional calculus), and axially monogenic functions (leading to the -functional calculus). By applying the respective product rule, we establish the four different -versions of these functional calculi.
Paper Structure (4 sections, 25 theorems, 241 equations)

This paper contains 4 sections, 25 theorems, 241 equations.

Table of Contents

  1. Introduction
  2. Preliminary results on quaternionic function theory and operators
  3. The $\omega$-functional calculus for decaying functions
  4. The quaternionic $H^\infty$-functional calculi

Key Result

Lemma 2.4

Let $T:V\rightarrow V$ be some right linear operator with $\operatorname{dom}(T)$ being a two-sided linear subspace of $V$. Then there exist unique two-sided linear operators $T_i:V\rightarrow V$ with $\operatorname{dom}(T_i)=\operatorname{dom}(T)$, $i\in\{0,1,2,3\}$, such that The components are explicitly given by

Theorems & Definitions (63)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • proof
  • Definition 2.8
  • ...and 53 more