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Additive maps preserving r-nilpotent perturbation of scalars on \(B({\mathcal {H}})\)

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Abstract

Let \({\mathcal {H}}\), \({\mathcal {K}}\) be Hilbert spaces over \({\mathbb {F}}\) with \(\dim {\mathcal {H}}\ge 3\), where \({\mathbb {F}}\) is the real or complex field. Assume that \(\varphi :{B}({\mathcal {H}})\rightarrow {B}({\mathcal {K}})\) is an additive surjective map and \(r\ge 3\) is a positive integer. It is shown that \(\varphi \) is r-nilpotent perturbation of scalars preserving in both directions if and only if either \(\varphi (A)=cTAT^{-1}+g(A)I\) holds for every \(A\in {B}({\mathcal {H}})\); or \(\varphi (A)=cTA^{*}T^{-1}+g(A)I\) holds for every \(A\in {B}({\mathcal {H}})\), where \(0\not =c\in {{\mathbb {F}}}\), \(T:{\mathcal {H}}\rightarrow {\mathcal {K}}\) is a \(\tau \)-linear bijective map with \(\tau :{\mathbb {F}}\rightarrow {\mathbb {F}}\) an automorphism and g is an additive map from \( B({\mathcal {H}})\) into \({{\mathbb {F}}}\). As applications, for any integer \(k\ge 5\), additive k-commutativity preserving maps and general completely k-commutativity preserving maps on \({B}({\mathcal {H}})\) are characterized, respectively.

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References

  1. 1.

    Bai, Z., Hou, J.: Linear maps and additive maps that preserve operators annihilated by a polynomial. J. Math. Anal. Appl. 271, 139–154 (2002)

  2. 2.

    Bai, Z., Hou, J.: Additive maps preserving nilpotent operators or spectral radius. Acta Math. Sin. Engl. Ser. 21–5, 1167–1182 (2005)

  3. 3.

    Botta, P., Pierce, S., Watkins, W.: Linear transformations that preserve the nilpotent matrices. Pac. J. Math. 104, 39–46 (1983)

  4. 4.

    Chebotar, M., Ke, W., Lee, P.: On maps preseving square-zero matrices. J. Algebra 289, 421–445 (2005)

  5. 5.

    Hou, J., Cui, J.: Introduction to the Linear Maps on Operator Algebras. Science Press, Beijing (2002)

  6. 6.

    Hou, J., Qi, X.: Strong \(k\)-commutativity preservers on complex standard operator algebras. Linear Multilinear Algebra 66–5, 902–918 (2018)

  7. 7.

    Hou, J., Zhao, L.: Zero-product preserving additive maps on symmetric operator spaces and self-adjoint operator spaces. Linear Algebra Appl. 399, 235–244 (2005)

  8. 8.

    Li, C., Tsing, N.: Linear prserver problems: a brief introduction and some special techniques. Linear Algebra Appl. 162–164, 217–235 (1992)

  9. 9.

    Qi, X., Hou, J.: Strong 3-commutativity preserving maps on prime rings. Linear Multilinear Algebra 65–11, 1–17 (2016)

  10. 10.

    Šemrl, P.: Linear mappings preserving square-zero matrices. Bull. Aust. Math. Soc. 48, 365 (1993)

  11. 11.

    Šemrl, P.: Linear maps that preserver the nilpotent operators. Acta. Sci. Math. (Szeged) 61, 1523–1534 (1995)

  12. 12.

    Šemrl, P.: Linear mappings that preserve operators annihilated by a polynomial. J. Oper. Theory 36, 45–58 (1996)

  13. 13.

    Wu, J., Li, P., Lu, S.: Additive mappings that preserve rank one nilpotent operators. Linear Algebra Appl. 367, 213–224 (2003)

  14. 14.

    Zhang, T., Hou, J.: Additive maps preserving nilpotent perturbation of scalars. Acta Math. Sin. Engl. Ser. 35–3, 407–426 (2019)

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Acknowledgements

The authors wish to express their thanks to the referee(s) for many helpful comments. This work is partially supported by National Natural Science Foundation of China (Nos.11671006 and 11671294) and Outstanding Youth Foundation of Shanxi Province (No. 201701D211001).

Author information

Correspondence to Xiaofei Qi.

Additional information

Communicated by Takeaki Yamazaki.

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Zhang, T., Hou, J. & Qi, X. Additive maps preserving r-nilpotent perturbation of scalars on \(B({\mathcal {H}})\). Ann. Funct. Anal. (2020). https://doi.org/10.1007/s43034-020-00060-2

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Keywords

  • Preservers
  • r-nilpotent operators
  • Commutativity
  • Hilbert spaces

Mathematics Subject Classification

  • 47B49