Fredholm theory in quaternionic Banach algebras

TL;DR

This paper develops a comprehensive framework for Fredholm theory within quaternionic Banach algebras, extending spectral analysis and perturbation results to quaternionic operators and their spectra.

Contribution

It introduces the Fredholm S-spectrum in quaternionic Banach algebras, analyzes its behavior under addition and perturbation, and applies these results to quaternionic linear operators.

Findings

Fredholm S-spectrum of sums can be characterized

Perturbation results for the S-spectrum are established

Applications to Fredholm and Weyl S-spectra of quaternionic operators

Abstract

Muraleetharan and Thirulogasanthan in (J. Math. phys. 59, No. 10, 103506, 27p. (2018)) introduced the concept of Calkin Sspectrum of a bounded quaternionic linear operators. The study of this spectrum is establisched using the Fredholm operators theory. Motivated by this, we study the general framework of the Fredholm element with respect to a quaternionic Banach algebra homomorphism. First, we investigate the Fredholm S-spectrum of the sum of two elements in quaternionic Banach algebra by means of the Fredholm S-spectrum of the two elements. Next, we prove a perturbation result on this spectrum. We also study the boundary S-spectrum. As application, we investigate the Fredholm and Weyl S-spectra of bounded right quaternionic linear operators.