Year: 2026
Type: preprint
Abstract: In this article, we introduce and study the class $\mathcal{M}_r(H)$ of minimum attaining operators on a separable Hilbert space $H$ whose restrictions to every nonzero reducing subspace are minimum attaining. This class lies between the classes of absolutely minimum attaining operators and minimum attaining operators. We establish a spectral characterization of positive operators in $\mathcal{M}_r(H)$: a positive operator $T$ belongs to $\mathcal{M}_r(H)$ if and only if $T=\sum_{λ\inΛ}λP_λ$, where $Λ\subseteq[0,\|T\|]$ is a countable well-ordered set, $(P_λ)_{λ\inΛ}$ is a family of mutually orthogonal nonzero projections, and $\sum_{λ\inΛ}P_λ=I$ in the strong operator topology. We also establish a connection between $\mathcal{M}_r(H)$ and the class of operators attaining their norm on every nonzero reducing subspace. Further, we obtain stability results under suitable finite-rank and compact perturbations, and prove that $\mathcal{M}_r(H)$ is norm dense in $\mathcal{B}(H)$. We also establish the norm density of operators in this class having a nontrivial invariant half-space. Finally, we characterize normal operators in $\mathcal{M}_r(H)$, obtain a corresponding characterization for multiplication operators on $L^2(\mathbb{T})$, and characterize the condition $\mathcal{R}_T=\mathcal{R}_{|T|}$ in terms of the von Neumann algebra generated by $|T|$, showing that this condition forces $T$ to be normal.
Source: arXiv (Cornell University)
Authors: Puspendu, Nag, Ramesh, Golla
Language: English
Topic: Advanced Banach Space Theory
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7203871769