Weyl's Theorem for Bounded Self-Adjoint Operators
Abstract
Let T be a bounded self-adjoint operator on a separable infinite-dimensional right quaternionic Hilbert space, and let sigdS(T) be the set of right eigenvalues of finite type. We prove that Weyl's theorem holds for T: sigeS(T)=sigS(T)∖sigdS(T), which completes the inclusion obtained in the general case. The proof proceeds through a single three-way criterion: a point of sigS(T) lies outside sigeS(T), and equally lies in sigdS(T), exactly when it is isolated with a spectral atom of finite rank. The natural route through Riesz projections is not available as it stands, because a Riesz projection need not be orthogonal; we show that for a self-adjoint operator it in fact is, the Riesz projection of an isolated spectral part coinciding with the corresponding spectral projection, so that the algebraic and geometric multiplicities agree. A 2×2 quaternionic matrix shows that this last identity fails without self-adjointness, and a normal operator with genuinely spherical S-spectrum shows what has to change in the normal case.
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BibTeX
@misc{baloudi2026weyls,
title = {{Weyl's Theorem for Bounded Self-Adjoint Operators}},
author = {Hatem Baloudi and Mohamed Ali Dbeibia and Kamel Mahfoudhi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23682},
eprint = {2609.23682},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23682},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Weyl's Theorem for Bounded Self-Adjoint Operators
AU - Hatem Baloudi
AU - Mohamed Ali Dbeibia
AU - Kamel Mahfoudhi
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23682
UR - https://arxiv.org/abs/2609.23682
AB - Let T be a bounded self-adjoint operator on a separable infinite-dimensional right quaternionic Hilbert space, and let sigdS(T) be the set of right eigenvalues of finite type. We prove that Weyl's theorem holds for T: sigeS(T)=sigS(T)∖sigdS(T), which completes the inclusion obtained in the general case. The proof proceeds through a single three-way criterion: a point of sigS(T) lies outside sigeS(T), and equally lies in sigdS(T), exactly when it is isolated with a spectral atom of finite rank. The natural route through Riesz projections is not available as it stands, because a Riesz projection need not be orthogonal; we show that for a self-adjoint operator it in fact is, the Riesz projection of an isolated spectral part coinciding with the corresponding spectral projection, so that the algebraic and geometric multiplicities agree. A 2×2 quaternionic matrix shows that this last identity fails without self-adjointness, and a normal operator with genuinely spherical S-spectrum shows what has to change in the normal case.
ER -