On symmetricity of the norm derivatives orthogonality in operator spaces
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| author | Ghosh, Souvik Paul, Kallol Sain, Debmalya |
| author_facet | Ghosh, Souvik Paul, Kallol Sain, Debmalya |
| contents | We investigate $ρ$-orthogonality and its local symmetry in the space of bounded linear operators. A characterization of Hilbert space operators with symmetric numerical range is established in terms of $ρ$-orthogonality. Further, we provide characterizations of $ρ$-left and $ρ$-right symmetric operators on finite-dimensional Hilbert spaces. In the two-dimensional real case, we show that the only nonzero $ρ$-left (or $ρ$-right) symmetric operators are scalar multiples of orthogonal matrices. However, in any finite-dimensional Hilbert space of dimension greater than two, an operator is $ρ$-left (or $ρ$-right) symmetric if and only if it is the zero operator. For infinite-dimensional spaces, we show that within a large class of operators, the zero operator remains the only example of $ρ$-left and $ρ$-right symmetric operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_11208 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On symmetricity of the norm derivatives orthogonality in operator spaces Ghosh, Souvik Paul, Kallol Sain, Debmalya Functional Analysis Primary: 46B20, Secondary: 47L05, 47A12 We investigate $ρ$-orthogonality and its local symmetry in the space of bounded linear operators. A characterization of Hilbert space operators with symmetric numerical range is established in terms of $ρ$-orthogonality. Further, we provide characterizations of $ρ$-left and $ρ$-right symmetric operators on finite-dimensional Hilbert spaces. In the two-dimensional real case, we show that the only nonzero $ρ$-left (or $ρ$-right) symmetric operators are scalar multiples of orthogonal matrices. However, in any finite-dimensional Hilbert space of dimension greater than two, an operator is $ρ$-left (or $ρ$-right) symmetric if and only if it is the zero operator. For infinite-dimensional spaces, we show that within a large class of operators, the zero operator remains the only example of $ρ$-left and $ρ$-right symmetric operators. |
| title | On symmetricity of the norm derivatives orthogonality in operator spaces |
| topic | Functional Analysis Primary: 46B20, Secondary: 47L05, 47A12 |
| url | https://arxiv.org/abs/2512.11208 |