Singularity preservers on the set of bounded observables (Q781715)

Let \(B(H)\) be the algebra of all bounded linear operators on a complex Hilbert space \(H\) of dimension at least 2, and \(B_s(H)\) be the set of all bounded selfadjoint operators on \(H\). Let \({\mathcal{SA}}_s(H)\) (resp., \({\mathcal{IA}}_s(H)\)) be the class of all singular (resp., invertible) algebraic operators in \(B_s(H)\). The authors first show that, if \(\Phi:B_s(H)\to B_s(H)\) is a unital additive surjective map, then the following assertions are equivalent. \begin{enumerate} \item For any operator \(S\in B_s(H)\), we have \(S\in{\mathcal{SA}}_s(H)\) if and only if \(\Phi(S)\in {\mathcal{SA}}_s(H)\). \item For any operator \(S\in B_s(H)\), we have \(S\in{\mathcal{IA}}_s(H)\) if and only if \(\Phi(S)\in{\mathcal{IA}}_s(H)\). \end{enumerate} Then they characterize all unital additive surjective maps on \(B_s(H)\) that satisfy any of the above two equivalent statements. They also present some auxiliary results that are used in the proof of the main results. Given a positive integer \(k\), the first result characterizes nonzero operators in \(B_s(H)\) that are of rank at most \(k\). The second one states that two operators \(T\) and \(S\) are in \({\mathcal{IA}}_s(H)\) provided that, for every rank one operator \(F\in B_s(H)\), one has \(S-F\in {\mathcal{IA}}_s(H)\) if and only if \(T-F\in {\mathcal{IA}}_s(H)\). Some final examples and remarks are stated in the last section.