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).