Quaternionic resolvent equation and series expansion of the $\mathcal{S}$-resolvent operator
Spectral Theory
2024-02-02 v1 Complex Variables
Functional Analysis
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.
Keywords
Cite
@article{arxiv.2402.00511,
title = {Quaternionic resolvent equation and series expansion of the $\mathcal{S}$-resolvent operator},
author = {Riccardo Ghiloni and Vincenzo Recupero},
journal= {arXiv preprint arXiv:2402.00511},
year = {2024}
}