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Singularity preservers on the set of bounded observables

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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).

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Acknowledgements

We are grateful to the referee whose precise comments as well as corrections shortened and polished some of the arguments. The third author is a fellow of the Iranian Academy of Sciences as well as a member of the Iranian National Elite Foundation; he wishes to thank these institutes for their general support.

Author information

Correspondence to Mina Jamshidi.

Additional information

Communicated by Qing-Wen Wang.

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Cite this article

Nayeri, M.D., Jamshidi, M. & Radjabalipour, M. Singularity preservers on the set of bounded observables. Ann. Funct. Anal. (2020). https://doi.org/10.1007/s43034-019-00050-z

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Keywords

  • Additive preserver problem
  • Selfadjoint operators
  • Algebraic singular operator

Mathematics Subject Classification

  • 15A86
  • 47B49
  • 47B15