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arXiv · 2512.11208

On symmetricity of the norm derivatives orthogonality in operator spaces

Souvik Ghosh·Kallol Paul·Debmalya Sain
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Abstract

We investigate $\rho$-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 $\rho$-orthogonality. Further, we provide characterizations of $\rho$-left and $\rho$-right symmetric operators on finite-dimensional Hilbert spaces. In the two-dimensional real case, we show that the only nonzero $\rho$-left (or $\rho$-right) symmetric operators are scalar multiples of orthogonal matrices. However, in any finite-dimensional Hilbert space of dimension greater than two, an operator is $\rho$-left (or $\rho$-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 $\rho$-left and $\rho$-right symmetric operators.

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Souvik Ghosh, Kallol Paul, Debmalya Sain. 2025-12-12. On symmetricity of the norm derivatives orthogonality in operator spaces. https://arxiv.org/abs/2512.11208

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