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

TL;DR

This paper investigates the essential spectrum of quaternionic linear operators, introducing the quaternionic Riesz projection and analyzing spectral properties in quaternionic Hilbert spaces.

Contribution

It introduces the quaternionic Riesz projection and studies the essential S-spectrum, providing new insights into quaternionic operator spectral theory.

Findings

Weyl and essential S-spectra do not contain eigenvalues of finite type.

Boundary of the Weyl S-spectrum is characterized.

Spectral theorem for the essential S-spectrum is described.

Abstract

This paper is devoted to the investigation of the Weyl and the essential $S-$spectrum 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 introduced and studied. In particular, it is shown that the Weyl and the essential $S-$spectra does not contains eigenvalues of finite type. We also describe the boundary of the Weyl $S-$spectrum and the particular case of the spectral theorem of the essential $S-$spectrum.