Fredholm operators and essential S-spectrum in the quaternionic setting

TL;DR

This paper investigates the spectral theory of bounded right linear operators in quaternionic Hilbert spaces, focusing on the approximate S-point spectrum, Fredholm operators, and the essential S-spectrum, including invariance properties and spectral characterizations.

Contribution

It develops the theory of Fredholm operators and the essential S-spectrum in quaternionic Hilbert spaces, extending spectral analysis in this non-commutative setting.

Findings

Fredholm index is invariant under small norm and compact perturbations.

Characterization of the S-spectrum via the essential S-spectrum and Fredholm operators.

Development of the theory of essential S-spectrum in quaternionic Hilbert spaces.

Abstract

For bounded right linear operators, in a right quaternionic Hilbert space with a left multiplication defined on it, we study the approximate $S$-point spectrum. In the same Hilbert space, then we study the Fredholm operators and the Fredholm index. In particular, we prove the invariance of the Fredholm index under small norm operator and compact operator perturbations. Finally, in association with Fredholm operators, we develop the theory of essential S-spectrum. We also characterize the $S$-spectrum in terms of the essential S-spectrum and Fredholm operators. In the sequel we study left and right S-spectrums as needed for the development of the theory presented in this note.