On Absolutely norm (minimum) attaining $2\times 2$ block operator matrix
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911057896800256 |
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| author | Nag, Puspendu Golla, Ramesh |
| author_facet | Nag, Puspendu Golla, Ramesh |
| contents | In this article, we study absolutely norm attaining operators ($\mathcal{AN}$-operators, in short), that is, operators that attain their norm on every non-zero closed subspace of a Hilbert space. Our focus is primarily on positive $2\times2$ block operator matrices in Hilbert spaces. Subsequently, we examine the analogous problem for operators that attain their minimum modulus on every nonzero closed subspace; these are referred to as absolutely minimum attaining operators (or $\mathcal{AM}$-operators, in short). We provide conditions under which these operators belong to the operator norm closure of the above two classes. In addition, we give a characterization of idempotent operators that fall into these three classes. Finally, we illustrate our results through examples that involve concrete operators. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_11148 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Absolutely norm (minimum) attaining $2\times 2$ block operator matrix Nag, Puspendu Golla, Ramesh Functional Analysis 47A08, 47A53, 47B07, 47B35 In this article, we study absolutely norm attaining operators ($\mathcal{AN}$-operators, in short), that is, operators that attain their norm on every non-zero closed subspace of a Hilbert space. Our focus is primarily on positive $2\times2$ block operator matrices in Hilbert spaces. Subsequently, we examine the analogous problem for operators that attain their minimum modulus on every nonzero closed subspace; these are referred to as absolutely minimum attaining operators (or $\mathcal{AM}$-operators, in short). We provide conditions under which these operators belong to the operator norm closure of the above two classes. In addition, we give a characterization of idempotent operators that fall into these three classes. Finally, we illustrate our results through examples that involve concrete operators. |
| title | On Absolutely norm (minimum) attaining $2\times 2$ block operator matrix |
| topic | Functional Analysis 47A08, 47A53, 47B07, 47B35 |
| url | https://arxiv.org/abs/2507.11148 |