Fredholm theory in quaternionic Banach algebras

Abstract

Muraleetharan and Thirulogasanthar (J. Math. Phys. 2018;59(10):103506, 27p.) introduced the concept of the Calkin S-spectrum of a bounded quaternionic linear operator. The study of this spectrum is established using Fredholm operator 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 a 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 an application, we investigate the Fredholm and Weyl S-spectra of bounded right quaternionic linear operators.