Fredholm operators and essential S-spectrum in the quaternionic setting

Abstract

In a right quaternionic Hilbert space, for a bounded right linear operator, the Kato S-spectrum is introduced and studied to a certain extent. In particular, it is shown that the Kato S-spectrum is a non-empty compact subset of the S-spectrum and it contains the boundary of the S-spectrum. Using right-slice regular functions, local Sspectrum, at a point of a right quaternionic Hilbert space, and the local spectral subsets are introduced and studied. The S-surjectivity spectrum and its connections to the Kato S-spectrum, approximate S-point spectrum and local S-spectrum are investigated. The generalized Kato S-spectrum is introduced and it is shown that the generalized Kato S-spectrum is a compact subset of the S-spectrum.

Weyl and Browder S-spectra in a right quaternionic Hilbert space

“…In order to study the Weyl and Browder operators and Weyl and Browder S-spectra we need some results regarding the Fredholm operators. We borrow the materials of this section from [18] as needed for the development of the manuscript. For an enhanced explanation we refer the reader to [18].…”

“…Also in the study of quaternionic Weyl and Browder operators and S-spectra results regarding quaternionic Fredholm operators and quaternionic essential S-spectrum are involved. Materials regarding these two topics are heavily borrowed from the recent paper [18] as needed here.…”

“…In section 3.2 we deal with the right S-spectrum, left S-spectrum, S-spectrum and its major partitions. In section 4 we recall some facts about the Fredholm operators and its index for a bounded quaternionic right linear operator from [18]. In section 5, from [18] we recall results about the essential S-spectrum as needed.…”

“…In section 4 we recall some facts about the Fredholm operators and its index for a bounded quaternionic right linear operator from [18]. In section 5, from [18] we recall results about the essential S-spectrum as needed. We also prove certain results which are omitted from [18].…”

“…In section 5, from [18] we recall results about the essential S-spectrum as needed. We also prove certain results which are omitted from [18]. In section 6 we introduce quaternionic Weyl operators and Weyl S-spectrum.…”