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).

中文翻译：

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有界可观察集上的奇点保护器

令 $$B_s(H)$$ 表示作用于维度为 $$\ge 2$$ 的可分离复 Hilbert 空间 H 上的所有有界自伴随算子的集合。另外，让 $${\mathcal {S}}{\mathcal {A}}_s(H)$$（特别是 $${\mathcal {I}}{\mathcal {A}}_s(H)$$ ) 表示 $$B_s(H)$$ 中所有奇异（可逆）代数算子的类。假设 $${\varPhi }:B_s(H)\rightarrow B_s(H)$$ 是一个单位加性满射映射，使得 $${\varPhi }({\mathcal {S}}{\mathcal {A}}_s (H))={\mathcal {S}}{\mathcal {A}}_s(H)$$ (resp. $${\varPhi }({\mathcal {I}}{\mathcal {A}}_s (H))={\mathcal {I}}{\mathcal {A}}_s(H)$$)。然后 $${\varPhi }(T)=\tau T\tau ^{-1}~\forall T\in B_s(H)$$，其中 $$\tau$$ 是酉算符或反酉算符。特别是，$${\varPhi }$$ 保留了 Molnar 感兴趣的 $$B_s(H)$$ 上的 $$\le$$ 顺序 (J Math Phys 42(12):5904–5909, 2001) .