On the symmetricity of norm-derivative orthogonality in operator spaces

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 about the origin 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 multiple 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.