Linear maps preserving families of C-symmetric operators

Abstract

Let $\cH$ be a separable complex Hilbert space with $\dim\cH\ge3$. We characterize the bounded bijective complex-linear maps $T$ on $\BH$ for which there exists a bijection $\psi$ of the set of conjugations,preserving commutativity in both directions, such that $T$ maps the space of $C$-symmetric operators onto the space of $\psi (C)$-symmetric operators for every conjugation $C$. We show that this condition is equivalent to the existence of a bijection $\phi$ of the set of all orthonormal bases such that $T$ maps the set of all $(e_n)$-diagonal operators onto the set of all $\phi ((e_n))$-diagonal operators. We also characterize the corresponding preservers in dimension two using Pauli coordinates, without assuming the commutativity condition on $\psi$.