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Singularity preservers on the set of bounded observables

  • Maryam D. Nayeri
  • Mina JamshidiEmail author
  • Mehdi Radjabalipour
Original Paper

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).

Keywords

Additive preserver problem Selfadjoint operators Algebraic singular operator 

Mathematics Subject Classification

15A86 47B49 47B15 

Notes

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.

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Copyright information

© Tusi Mathematical Research Group (TMRG) 2020

Authors and Affiliations

  1. 1.Department of Applied MathematicsGraduate University of Advanced TechnologyKermanIran
  2. 2.Department of MathematicsErfan Institute of Higher EducationKermanIran

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