发表机构
School of Mathematics, Jilin University（吉林大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究无限维希尔伯特空间上约化子空间范数达到算子的类$\beta(H)$，证明其在算子范数下稠密，并引入子类$\beta_0(H)$，刻画其有限秩与紧扰动稳定性，阐明谱结构。
AI 中文摘要
对于无限维希尔伯特空间上的有界线性算子，我们考虑算子类$\beta(H)$，该类中的算子在所有非零约化子空间上的限制均能达到其范数。我们证明$\beta(H)$在算子范数意义下于$\mathcal{B}(H)$中稠密。由于$\beta(H)$在任意紧扰动下不稳定，我们引入一个自然子类$\beta_0(H)$并研究其扰动性质。我们证明$\beta_0(H)$在有限秩扰动下稳定，并给出保持$\beta_0(H)$成员资格的所有紧扰动的完整刻画。我们还确定了$\beta_0(H)$中最大的紧扰动不变子集。这些结果阐明了约化子空间上范数达到算子的谱结构与稳定性。
英文摘要
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 $\mathcal{B}(H)$ 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.
