Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The $H^\infty$-functional calculus for bisectorial Clifford operators

Francesco Mantovani, Peter Schlosser

TL;DR

This work extends functional calculus to unbounded bisectorial Clifford operators by developing an $\omega$-functional calculus for functions with decay at $0$ and $\infty$, then extending to finite limits and finally constructing an $H^ ablafty$-calculus for regularizable functions. The three-step framework enables polynomial, rational, and product rules within the Clifford setting, and establishes a spectral mapping relation in the extended regime. The results rely on slice hyperholomorphic function theory, the $S$-spectrum, and the $S$-resolvent framework, bridging noncommutative Clifford analysis with operator theory. This provides a robust toolkit for evolution equations and spectral problems in higher-dimensional Clifford contexts, including gradient-type operators and other fully Clifford operators.

Abstract

The aim of this article is to introduce the H-infinity functional calculus for unbounded bisectorial operators in a Clifford module over the algebra R_n. While recent studies have focused on bounded operators or unbounded paravector operators, we now investigate unbounded fully Clifford operators and define polynomially growing functions of them. We first generate the omega-functional calculus for functions that exhibit an appropriate decay at zero and at infinity. We then extend to functions with a finite value at zero and at infinity. Finally, using a subsequent regularization procedure, we can define the H-infinity functional calculus for the class of regularizable functions, which in particular include functions with polynomial growth at infinity and, if T is injective, also functions with polynomial growth at zero.

The $H^\infty$-functional calculus for bisectorial Clifford operators

TL;DR

This work extends functional calculus to unbounded bisectorial Clifford operators by developing an -functional calculus for functions with decay at and , then extending to finite limits and finally constructing an -calculus for regularizable functions. The three-step framework enables polynomial, rational, and product rules within the Clifford setting, and establishes a spectral mapping relation in the extended regime. The results rely on slice hyperholomorphic function theory, the -spectrum, and the -resolvent framework, bridging noncommutative Clifford analysis with operator theory. This provides a robust toolkit for evolution equations and spectral problems in higher-dimensional Clifford contexts, including gradient-type operators and other fully Clifford operators.

Abstract

The aim of this article is to introduce the H-infinity functional calculus for unbounded bisectorial operators in a Clifford module over the algebra R_n. While recent studies have focused on bounded operators or unbounded paravector operators, we now investigate unbounded fully Clifford operators and define polynomially growing functions of them. We first generate the omega-functional calculus for functions that exhibit an appropriate decay at zero and at infinity. We then extend to functions with a finite value at zero and at infinity. Finally, using a subsequent regularization procedure, we can define the H-infinity functional calculus for the class of regularizable functions, which in particular include functions with polynomial growth at infinity and, if T is injective, also functions with polynomial growth at zero.
Paper Structure (7 sections, 35 theorems, 288 equations)

This paper contains 7 sections, 35 theorems, 288 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on Clifford Algebras and Clifford modules
  3. Slice hyperholomorphic functions
  4. Clifford modules
  5. The $\omega$-functional calculus
  6. Extended $\omega$-functional calculus
  7. The $H^\infty$-functional calculus

Key Result

Theorem 2.4

Let $U\subseteq\mathbb{R}^{n+1}$ be open, axially symmetric and $f\in\mathcal{SH}_L(U)$ (resp. $f\in\mathcal{SH}_R(U)$). Then for every $p\in U$ there holds In these integrals $J\in\mathbb{S}$ is arbitrary and $U'$ is any subset with $\overline{U'}\subseteq U$ and where the boundary $\partial U'\cap\mathbb{C}_J$ consists of finitely many continuously differentiable curves.

Theorems & Definitions (90)

  • Definition 2.1: Slice hyperholomorphic functions
  • Remark 2.2
  • Definition 2.3
  • Theorem 2.4: Cauchy integral formula
  • Definition 2.5: Closed operators
  • Definition 2.6: Bounded operators
  • Definition 2.7: $S$-spectrum
  • Remark 2.8
  • Remark 2.9
  • Remark 2.10
  • ...and 80 more