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Slice hyperholomorphicity of the $S$-resolvent operators and boundary conditions

Francesco Mantovani

TL;DR

This work extends slice hyperholomorphic spectral theory to include boundary conditions by introducing the $S$-spectrum with boundary data via a subspace $B$ and the restricted pseudo-resolvent $Q_{s,B}[T]$. It establishes the openness and $C^1$-dependence of $Q_{s,B}[T]^{-1}$ and studies the continuity and analyticity of the boundary-modified $S$-resolvent operators $S_{L,B}^{-1}$ and $S_{R,B}^{-1}$, revealing that holomorphy hinges on the commutativity between $T$ and $Q_{s,B}[T]^{-1}$. The paper provides general criteria for when the Cauchy-Riemann equations hold in this boundary setting, including a key result: if $\mathrm{dom}(T^2)\subset B$, then full slice hyperholomorphicity is recovered and a generalized $S$-resolvent equation holds with a commutator term. These findings pave the way for boundary-value spectral analysis of Clifford-operator PDEs and have potential impact on gradient, Dirac, and Dirac-type operator theories, as well as on related functional calculi.

Abstract

The foundation of spectral theory on the $S$-spectrum can be traced back to the quaternionic framework of quantum mechanics. The concept of $S$-spectrum for quaternionic operators emerged as the natural spectrum in slice hyperholomorphic functional calculi, known as the $S$-functional calculus and also utilized in the quaternionic spectral theorem. This spectral theory extends to Clifford operators. A key distinction from classical complex spectral theory lies in the definition of the $S$-spectrum, which is second order in the operator $T$, and in the $S$-resolvent operators that turns out to be the product of two different operators. This study delves into the analyticity of the $S$-resolvent operators under specified boundary conditions for the $S$-spectral problem. The spectral theory on the $S$-spectrum also provides deeper insights into classical spectral theory.

Slice hyperholomorphicity of the $S$-resolvent operators and boundary conditions

TL;DR

This work extends slice hyperholomorphic spectral theory to include boundary conditions by introducing the -spectrum with boundary data via a subspace and the restricted pseudo-resolvent . It establishes the openness and -dependence of and studies the continuity and analyticity of the boundary-modified -resolvent operators and , revealing that holomorphy hinges on the commutativity between and . The paper provides general criteria for when the Cauchy-Riemann equations hold in this boundary setting, including a key result: if , then full slice hyperholomorphicity is recovered and a generalized -resolvent equation holds with a commutator term. These findings pave the way for boundary-value spectral analysis of Clifford-operator PDEs and have potential impact on gradient, Dirac, and Dirac-type operator theories, as well as on related functional calculi.

Abstract

The foundation of spectral theory on the -spectrum can be traced back to the quaternionic framework of quantum mechanics. The concept of -spectrum for quaternionic operators emerged as the natural spectrum in slice hyperholomorphic functional calculi, known as the -functional calculus and also utilized in the quaternionic spectral theorem. This spectral theory extends to Clifford operators. A key distinction from classical complex spectral theory lies in the definition of the -spectrum, which is second order in the operator , and in the -resolvent operators that turns out to be the product of two different operators. This study delves into the analyticity of the -resolvent operators under specified boundary conditions for the -spectral problem. The spectral theory on the -spectrum also provides deeper insights into classical spectral theory.
Paper Structure (6 sections, 11 theorems, 121 equations)

This paper contains 6 sections, 11 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on Clifford Algebras and Clifford modules
  3. The $S$-spectrum with boundary conditions
  4. Continuity of the $S$-resolvent operators with boundary conditions
  5. Analyticity of the $S$-resolvent operators with boundary conditions
  6. Concluding remarks

Key Result

Proposition 3.4

Let $T:\mathop{\mathrm{dom}}\nolimits(T)\subseteq V \to V$ be a right-linear operator. Let $B \subseteq \mathop{\mathrm{dom}}\nolimits(T)$ be a right submodule and $s \in \rho_{S,B}(T)$. Then

Theorems & Definitions (24)

  • Definition 2.1: Slice hyperholomorphic functions
  • Definition 2.2: $S$-Spectrum
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.4
  • proof
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • ...and 14 more