Linear maps on symmetric matrix spaces preserving inverses of matrices (Q5920370)

Let \(F\) be a field of characteristic different from two, and let \(S_n(F)\subseteq M_n(F)\) be a subset of \(n\)-by-\(n\) symmetric matrices. It is shown that a linear map \(\Phi:S_n(F)\to M_n(F)\) preserves inverses (i.e., \(\Phi(X^{-1})=\Phi(X)^{-1}\) for invertible \(X\)) if and only if \(\Phi(X)=\varepsilon PXP^{-1}\) for some \(\varepsilon=\pm1\). As a consequence, the author also classifies linear maps that preserve inverses from \(S_n(F)\) to itself. Reviewer's remark: A related problem for additive maps that preserve inverses on Banach algebras can be found on p. 40 of \textit{M. Brešar} [Derivations, homomorphisms, and related mappings of rings and Banach algebras, Ph.D. Thesis, Univ. Ljubljana 1990 (in Slovene)]. It is proven that such a map is a Jordan triple product homomorphism. The proof can be modified to work also for additive maps on symmetric matrices, provided the characteristic of the underlying field is either zero or big enough.