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Quaternionic resolvent equation and series expansion of the $\mathcal{S}$-resolvent operator

Riccardo Ghiloni, Vincenzo Recupero

Abstract

In the present paper, we prove a resolvent equation for the $\mathcal{S}$-resolvent operator in the quaternionic framework. Exploiting this resolvent equation, we find a series expansion for the $\mathcal{S}$-resolvent operator in an open neighborhood of any given quaternion belonging to the $\mathcal{S}$-resolvent set. Some consequences of the series expansion are deduced. In particular, we describe a property of the geometry of the $\mathcal{S}$-resolvent set in terms of the Cassini pseudo-metric on quaternions. The concept of vector-valued real analytic function of several variables plays a crucial role in the proof of the mentioned series expansion for the $\mathcal{S}$-resolvent operator.

Quaternionic resolvent equation and series expansion of the $\mathcal{S}$-resolvent operator

Abstract

In the present paper, we prove a resolvent equation for the -resolvent operator in the quaternionic framework. Exploiting this resolvent equation, we find a series expansion for the -resolvent operator in an open neighborhood of any given quaternion belonging to the -resolvent set. Some consequences of the series expansion are deduced. In particular, we describe a property of the geometry of the -resolvent set in terms of the Cassini pseudo-metric on quaternions. The concept of vector-valued real analytic function of several variables plays a crucial role in the proof of the mentioned series expansion for the -resolvent operator.
Paper Structure (5 sections, 19 theorems, 108 equations)

This paper contains 5 sections, 19 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and main results
  3. Proof of Theorem \ref{['T:resolv.eq.']}
  4. Proof of Theorem \ref{['T:main']} and its corollaries
  5. Appendix: Vector-valued analytic functions

Key Result

Theorem 2.8

Assume that X holds. If $D(\mathsf{A})$ is a right $\mathbb{H}$-submodule of $X$ and $\mathsf{A} : D(\mathsf{A}) \longrightarrow X$ is a closed right linear operator, then $\mathsf{S}_q^{{}^{-1}}(\mathsf{A}) \in \mathscr{L}^\textsl{r}(X)$ for every $q\in\rho_\mathpzc{S}(\mathsf{A})$ and

Theorems & Definitions (50)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.8: Resolvent equation
  • Theorem 2.9: Series expansion of the $\mathcal{S}$-resolvent operator
  • Corollary 2.10
  • ...and 40 more