On symmetricity of the norm derivatives orthogonality in operator spaces

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.