Narrow Results

The paper aims to extend the concept of Fredholm, Weyl and Jeribi essential spectra in the quaternionic setting. Furthermore, some properties and stability of the corresponding spectra of Fredholm and Weyl operators have been investigated in this setting. To achieve the goal, a characterization of the sum of two invariant bounded linear operators has been obtained in order to explore various properties of the Fredholm operator and Weyl operator under some assumptions in quaternionic setting. Also, various sequential properties of the pseudo-resolvent operator, right quaternionic linear operator, Weyl operator, Weyl S-spectrum, Jeribi essential S-spectrum and some properties of $2\times 2$ block operator matrices have been discussed. The spectral mapping theorem of essential S-spectrum, Weyl S-spectrum and Jeribi essential S-spectrum for self-adjoint operators has been established. A characterization of the essential S-spectrum and Weyl S-spectrum of the sum of two bounded linear operators concludes this investigation.

In this paper, we study the notions of Browder operator and Browder S-spectrum of bounded right linear operator defined over the right quaternionic Hilbert space. Some properties of Browder operator and stability of the ascent, descent and Browder S-spectrum have been investigated in the right quaternionic setting. We also characterize the property of invariant Browder operators and study the spectral mapping theorem of Browder S-spectrum for self-adjoint operators in quaternionic setting. This investigation concludes by exploring the Browder S-spectrum of the sum of two bounded linear operators.