Singularity preservers on the set of bounded observables

Abstract

Let $B_s(H)$ denote the set of all bounded selfadjoint operators acting on a separable complex Hilbert space H of dimension $\ge 2$. Also, let ${\mathcal {S}}{\mathcal {A}}_s(H)$ (esp. ${\mathcal {I}}{\mathcal {A}}_s(H)$) denote the class of all singular (resp. invertible) algebraic operators in $B_s(H)$. Assume ${\varPhi }:B_s(H)\rightarrow B_s(H)$ is a unital additive surjective map such that ${\varPhi }({\mathcal {S}}{\mathcal {A}}_s(H))={\mathcal {S}}{\mathcal {A}}_s(H)$ (resp. ${\varPhi }({\mathcal {I}}{\mathcal {A}}_s(H))={\mathcal {I}}{\mathcal {A}}_s(H)$). Then ${\varPhi }(T)=\tau T\tau ^{-1}~\forall T\in B_s(H)$, where $\tau$ is a unitary or an antiunitary operator. In particular, ${\varPhi }$ preserves the order $\le$ on $B_s(H)$ which was of interest to Molnar (J Math Phys 42(12):5904–5909, 2001).