Linear maps preserving $C$-symmetry operators

Let $H$ be a separable complex Hilbert space. In this paper, we study a continuous bijective linear map $T$ on the algebra of all bounded linear operators on $H$. We characterize the map $T$ that maps the set of all $C$-symmetric operators onto the set of all $ψ(C)$-symmetric operators, where $ψ$ is a commutativity-preserving bijection on the set of all conjugations. Furthermore, we show that this condition is equivalent to the existence of a bijection $\varphi$ on the set of all orthonormal bases of $H$ such that $T$ maps the set of all $\{e_n\}$-diagonal operators onto the set of all $\varphi(\{e_n\})$-diagonal operators.