Bijections preserving invertibility of differences of matrices on \(H_n\) (Q953228)

Let \(H_n\) be the space of all \(n \times n\) Hermitian matrices and \(\varphi: H_n \to H_n\) be a bijective mapping such that \(\varphi(A)-\varphi(B)\) is invertible if and only if \(A-B\) is invertible for all \(A, B \in H_n\). The author shows that there exist an invertible \(n \times n\) complex matrix \(T\) and \(C\in H_n\) such that either \(\varphi(A)=\varepsilon TAT^*+C\) or \(\varphi(A)=\varepsilon T\overline{A}T^*+C\), where \(\varepsilon=\pm 1\). For a related work, see \textit{H. Havlicek} and \textit{P. Šemrl} [Stud. Math. 174, No.~1, 99--109 (2006; Zbl 1107.47023)].