Singularity preservers on the set of bounded observables

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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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  1. 1.

    Matsumoto, K.: Self-adjoint operators as a real span of 5 projections. Jpn J. Math. 29, 291–294 (1984)

  2. 2.

    Molnar, L.: Order-automorphisms of the set of bounded observables. J. Math. Phys. 42(12), 5904–5909 (2001)

  3. 3.

    Molnar, L.: Selected preserver problems on algebraic structures of linear operators and on function spaces. Lecture Notes in Mathematics, 1895. Springer, Berlin (2007)

  4. 4.

    Molnar, L.: Order automorphisms on positive definite operators and a few applications. Linear Algebra Appl. 434, 2158–2169 (2011)

  5. 5.

    Nakamura, Y.: Any Hermitian matrix is a linear combination of four projections. Linear Algebra Appl. 61, 133–139 (1984)

  6. 6.

    Šemrl, P.: Symmetries on bounded observables: a unified approach based on adjacency preserving maps. Integr. Equ. Oper. Theory 72, 7–66 (2012)

  7. 7.

    Souilah, K.: On additive preservers of certain classes of algebraic operators. Extract. Math. 30(2), 207–220 (2015)

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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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Correspondence to Mina Jamshidi.

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Communicated by Qing-Wen Wang.

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Nayeri, M.D., Jamshidi, M. & Radjabalipour, M. Singularity preservers on the set of bounded observables. Ann. Funct. Anal. (2020).

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  • Additive preserver problem
  • Selfadjoint operators
  • Algebraic singular operator

Mathematics Subject Classification

  • 15A86
  • 47B49
  • 47B15