AI 中文总结
本文引入可分希尔伯特空间上最小达到算子的新子类$\boldsymbol{\textit{M}}_r(H)$，刻画其算子结构与谱性质，证明该类在算子范数下于$\boldsymbol{\textit{B}}(H)$中稠密，还得到其正规算子表示定理。
AI 中文摘要
本文引入并研究可分希尔伯特空间$H$上最小达到算子的新子类$\boldsymbol{\textit{M}}_r(H)$。该类包含绝对最小达到算子，且真包含于最小达到算子类中。我们建立$\boldsymbol{\textit{M}}_r(H)$中算子的若干结构与谱刻画，尤其通过谱表示刻画$\boldsymbol{\textit{M}}_r(H)$中的正算子。证明$\boldsymbol{\textit{M}}_r(H)$在算子范数下于$\boldsymbol{\textit{B}}(H)$中稠密，且具有非平凡不变半空间的$\boldsymbol{\textit{M}}_r(H)$中算子也在算子范数下于$\boldsymbol{\textit{B}}(H)$中稠密。进一步得到$\boldsymbol{\textit{M}}_r(H)$中正规算子的表示定理，并建立该类的附加结构性质。
英文摘要
In this article, we introduce and investigate a new subclass $\mathcal{M}_r(H)$ of minimum attaining operators on a separable Hilbert space $H$. This class contains the absolutely minimum attaining operators and is properly contained in the class of minimum attaining operators. We establish several structural and spectral characterizations of operators in $\mathcal{M}_r(H)$. In particular, we characterize positive operators in $\mathcal{M}_r(H)$ in terms of their spectral representations. We prove that $\mathcal{M}_r(H)$ is dense in $\mathcal{B}(H)$ in the operator norm and, moreover, that the operators in $\mathcal{M}_r(H)$ having a nontrivial invariant half-space are also dense in $\mathcal{B}(H)$ in the operator norm. We further obtain a representation theorem for normal operators in $\mathcal{M}_r(H)$ and establish additional structural properties of this class.
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