Riesz projection and essential $S$-spectrum in quaternionic setting

Abstract

This paper is devoted to the investigation of the Weyl and the essential S−spectra of a bounded right quaternionic linear operator in a right quaternionic Hilbert space. Using the quaternionic Riesz projection, the S−eigenvalue of finite type is both introduced and studied.In particular, we have shown that the Weyl and the essential S−spectra do not contain eigenvalues of finite type. We have also described the boundary of the Weyl S−spectrum and the particular case of the spectral theorem of the essential S−spectrum. Contents 1. Introduction 1 2. Mathematical preliminaries 3 2.1. Quaternions 3 2.2. Right quaternionic Hilbert space and operator 4 2.3. The quaternionic functional calculus 7 3. Riesz projection and essential spectrum 9 4. Some results on the Weyl S-spectrum 18 References 22