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The $H^\infty$-functional calculus for right slice hyperholomorphic functions and right linear Clifford operators

Fabrizio Colombo, Francesco Mantovani, Peter Schlosser

TL;DR

This work extends the $H^\infty$-functional calculus to right slice hyperholomorphic functions for injective bisectorial right-linear Clifford operators on Clifford modules. It defines the right calculus via $f(T)=\overline{(fe)(T)e(T)^{-1}}$, with a regularizer $e(s)=\frac{s^m}{(1+s^2)^m}$, and proves independence from the choice of $m$ by exploiting the decomposition $V=\ker(T)\oplus\overline{\operatorname{ran}}(T)$ and related density results. The calculus coherently connects with the existing left $H^\infty$-calculus for intrinsic functions and the right $\\omega$-calculus for decaying functions, while treating the right case as a multivalued operator in general. The results provide a consistent, fully developed right slice hyperholomorphic functional calculus in Clifford analysis, with potential applications to evolution equations and spectral theory in noncommutative settings.

Abstract

In 2016, the spectral theory on the $S$-spectrum was used to establish the $H^\infty$-functional calculus for quaternionic or Clifford operators. This calculus applies for example to sectorial or bisectorial right linear operators $T$ and left slice hyperholomorphic functions $f$ that can grow as polynomials. It relies on the product of the two operators $e(T)^{-1}$ and $(ef)(T)$, both defined via some underlying $S$-functional calculus (also called $ω$-functional calculus). For left slice holomorphic functions $f$ this definition does not depend on the choice of the regularizer function $e$. However, due to the non-commutative multiplication of Clifford numbers, it was unclear how to extend this definition to right slice hyperholomorphic functions. This paper addresses this significant unresolved issue and shows how right linear operators can possess the $H^\infty$-functional calculus also for right slice hyperholomorphic functions.

The $H^\infty$-functional calculus for right slice hyperholomorphic functions and right linear Clifford operators

TL;DR

This work extends the -functional calculus to right slice hyperholomorphic functions for injective bisectorial right-linear Clifford operators on Clifford modules. It defines the right calculus via , with a regularizer , and proves independence from the choice of by exploiting the decomposition and related density results. The calculus coherently connects with the existing left -calculus for intrinsic functions and the right -calculus for decaying functions, while treating the right case as a multivalued operator in general. The results provide a consistent, fully developed right slice hyperholomorphic functional calculus in Clifford analysis, with potential applications to evolution equations and spectral theory in noncommutative settings.

Abstract

In 2016, the spectral theory on the -spectrum was used to establish the -functional calculus for quaternionic or Clifford operators. This calculus applies for example to sectorial or bisectorial right linear operators and left slice hyperholomorphic functions that can grow as polynomials. It relies on the product of the two operators and , both defined via some underlying -functional calculus (also called -functional calculus). For left slice holomorphic functions this definition does not depend on the choice of the regularizer function . However, due to the non-commutative multiplication of Clifford numbers, it was unclear how to extend this definition to right slice hyperholomorphic functions. This paper addresses this significant unresolved issue and shows how right linear operators can possess the -functional calculus also for right slice hyperholomorphic functions.
Paper Structure (5 sections, 11 theorems, 142 equations)

This paper contains 5 sections, 11 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on Clifford Algebras and Clifford modules
  3. Properties of bisectorial operators
  4. The $H^\infty$-functional calculus right slice hyperholomorphic functions
  5. Concluding remarks

Key Result

Lemma 3.2

Let $T\in\mathcal{K}(V)$ be bisectorial of angle $\omega\in(0,\frac{\pi}{2})$. Then for every $\varphi\in(\omega,\frac{\pi}{2})$, and with the corresponding constant $C_\varphi$ from Eq_SL_estimate, there holds the norm estimates

Theorems & Definitions (32)

  • Definition 2.1: Slice hyperholomorphic functions
  • Definition 2.2: Multivalued operators
  • Definition 3.1: Bisectorial operators
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • ...and 22 more