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The Stability and Density of Norm Attaining Operators on Reducing Subspaces
Abstract
For a bounded linear operator on an infinite-dimensional Hilbert space, we consider the class β(H) of operators whose restrictions to all nonzero reducing subspaces attain their norms. We prove that β(H) is dense in ℬ(H) in the operator norm. Since β(H) is not stable under arbitrary compact perturbations, we introduce a natural subclass β0(H) and investigate its perturbation properties. We show that β0(H) is stable under finite-rank perturbations and obtain a complete characterization of those compact perturbations that preserve membership in β0(H). We also determine the maximal compactly perturbation-invariant subset of β0(H). These results clarify the spectral structure and stability of norm-attaining operators on reducing subspaces.
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BibTeX
@misc{su2026stability,
title = {{The Stability and Density of Norm Attaining Operators on Reducing Subspaces}},
author = {Zinan Su and Yuanhang Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37621},
eprint = {2609.37621},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37621},
}
RIS
TY - UNPB
TI - The Stability and Density of Norm Attaining Operators on Reducing Subspaces
AU - Zinan Su
AU - Yuanhang Zhang
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.37621
UR - https://arxiv.org/abs/2609.37621
AB - For a bounded linear operator on an infinite-dimensional Hilbert space, we consider the class β(H) of operators whose restrictions to all nonzero reducing subspaces attain their norms. We prove that β(H) is dense in ℬ(H) in the operator norm. Since β(H) is not stable under arbitrary compact perturbations, we introduce a natural subclass β_0(H) and investigate its perturbation properties. We show that β_0(H) is stable under finite-rank perturbations and obtain a complete characterization of those compact perturbations that preserve membership in β_0(H). We also determine the maximal compactly perturbation-invariant subset of β_0(H). These results clarify the spectral structure and stability of norm-attaining operators on reducing subspaces.
ER -