Minimum attaining operators on reducing subspaces: Spectral structure and density

Abstract

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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