On symmetricity of orthogonality in function spaces and space of operators on Banach spaces

We study symmetric points with respect to $({\rho }_{+})$-orthogonality, $({\rho }_{-})$-orthogonality and $\rho$-orthogonality in the space $C(K,\mathbb{X}),$ where $K$ is a perfectly normal, compact space and $\mathbb{X}$ is a Banach space. We characterize left symmetric points and right symmetric points in $C(K,\mathbb{X})$ with respect to $({\rho }_{+})$-orthogonality and $({\rho }_{-})$-orthogonality, separately. Furthermore, we provide necessary conditions for left symmetric and right symmetric points with respect to $\rho$-orthogonality. As an application of these results we also study these symmetric points in the space of operators defined on some special Banach spaces.