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PreprintCC BY 4.0arXiv

A note on the Browder S-spectrum of a bounded quaternionic operator

  • Mohamed Ali Dbeibia
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Abstract

Let A be a bounded operator on a separable right quaternionic Hilbert space and let CA be the set of compact operators commuting with A. We establish the formula sigb(A)=⋂K∈CA sS(A+K), a quaternionic analogue of the classical characterization of the complex Browder spectrum. The proof rests on a factorization lemma (Lemma~\ref{lem:fact}), a consequence of the Riesz--Schauder theorem and an ascent/descent argument, and on the characterization sigb(A)=sS(A)∖sd(A) in terms of S-eigenvalues of finite type, due to Arzini and Jaatit \cite{AJ26}, for which we give here an independent proof (Lemma~\ref{lem:A}). We also record two consequences of independent interest: the intersection defining sigb(A) may be restricted, without loss, to \emph{finite-rank} operators commuting with A (Corollary~\ref{cor:finite}); and, by a separate argument resting on the ascent/descent stability results of \cite{KD24}, the stronger pointwise statement sigb(A+K)=sigb(A) holds for a fixed K∈CA and arbitrary A∈B(V) (Theorem~\ref{thm:pointwise}), known in the complex case.

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Cite

BibTeXRIS
BibTeX
@misc{dbeibia2026note,
  title = {{A note on the Browder S-spectrum of a bounded quaternionic operator}},
  author = {Mohamed Ali Dbeibia},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.10616},
  eprint = {2609.10616},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.10616},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A note on the Browder S-spectrum of a bounded quaternionic operator
AU  - Mohamed Ali Dbeibia
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.10616
UR  - https://arxiv.org/abs/2609.10616
AB  - Let A be a bounded operator on a separable right quaternionic Hilbert space and let CA be the set of compact operators commuting with A. We establish the formula sigb(A)=⋂_K∈CA sS(A+K), a quaternionic analogue of the classical characterization of the complex Browder spectrum. The proof rests on a factorization lemma (Lemma~\ref{lem:fact}), a consequence of the Riesz--Schauder theorem and an ascent/descent argument, and on the characterization sigb(A)=sS(A)∖sd(A) in terms of S-eigenvalues of finite type, due to Arzini and Jaatit \cite{AJ26}, for which we give here an independent proof (Lemma~\ref{lem:A}). We also record two consequences of independent interest: the intersection defining sigb(A) may be restricted, without loss, to \emph{finite-rank} operators commuting with A (Corollary~\ref{cor:finite}); and, by a separate argument resting on the ascent/descent stability results of \cite{KD24}, the stronger pointwise statement sigb(A+K)=sigb(A) holds for a fixed K∈CA and arbitrary A∈B(V) (Theorem~\ref{thm:pointwise}), known in the complex case.
ER  -