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Higher Dimensional Numerical Range of \(\xi \)-Lie Products on Bound Self-adjoint Operators

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Abstract

Let H be a complex Hilbert space and \({{\mathcal {B}}}_s(H)\) the Jordan algebra of all bounded self-adjoint linear operators on H. Assume that k is any positive integer with \(k<\dim H\) and \(\xi \) is any complex number. In this paper, we first give some useful properties about k-dimensional numerical range of \(\xi \)-Lie product \([A,B]_\xi =AB-\xi BA\) for \(A,B\in {{\mathcal {B}}}_s(H)\); and then, based on these properties, all surjective maps preserving k-dimensional numerical ranges of \(\xi \)-Lie product on \({\mathcal B}_s(H)\) are completely characterized.

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References

  1. Bonsall, F.F., Duncan, J.: Studies in Functional Analysis-Numerical Ranges, Studies in Mathematics, vol. 21. MAA (1980)

  2. Chen, C.-Q., Lu, F.-Y.: Maps that preserve higher-dimensional numerical ranges with operator Jordan products. Linear Multilinear Algebra 65, 2530–2537 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Fillmore, P.A., William, J.P.: Some convexity theorems for matrices. Glasg. Math. J. 12, 110–117 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  4. Fos̆ner, A., Huang, Z.-J., Li, C.-K., Poon, Y.-T., Sze, N.-S.: Linear maps preserving the higher numerical ranges of tensor products of matrices. Linear Multilinear Algebra 62, 776–791 (2014)

  5. Gau, H.-L., Li, C.-K.: \(C^*\)-isomorphisms, Jordan isomorphisms, and numerical range preserving maps. Proc. Am. Math. Soc. 135, 2907–2914 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gustafson, K.E., Rao, D.K.M.: Numerical Range: The Field of Values of Linear Operators and Matrices. Springer, New york (1997)

    Book  Google Scholar 

  7. Halmos, P.R.: A Hilbert Space Problem Book, 2nd edn. Springer, New York (1982)

    Book  MATH  Google Scholar 

  8. He, K., Hou, J.-C., Dolinar, G., Kuzma, B.: Maps on self-adjoint operators preserving numerical range of products up to a factor. Acta. Math. Sin. (Chin. Ser.) 54, 925–932 (2011)

    MathSciNet  MATH  Google Scholar 

  9. Hou, J.-C., Li, C.-K., Qi, X.-F.: Numerical range of Lie product of operators. Integr. Equ. Oper. Theory 83, 497–516 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hou, J.-C., Di, Q.-H.: Maps preserving numerical ranges of operator products. Proc. Am. Math. Soc. 134, 1435–1446 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hou, J.-C., He, K.: Non-linear maps on self-adjoint operators preserving numerical radius and numerical range of Lie product. Linear Multilinear Algebra 64, 36–57 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Li, C.-K.: \(C\)-numerical ranges and \(C\)-numerical radii. Linear Multilinear Algebra 37, 51–82 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  13. Sun, S.-X., Qi, X.-F.: Higher dimensional numerical range of Lie products on self-adjoint operators. Banach J. Math. Anal. 16, 27 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  14. Wigner, E.P.: Gruppentheorie, pp. 251–254. Friedrich Vieweg und Sohn, Braunschweig (1931)

    MATH  Google Scholar 

  15. Wigner, E.P.: Group Theory, pp. 233–236. Academic Press Inc., New York (1959)

    Google Scholar 

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Acknowledgements

The authors wish to give their thanks to the referees for their helpful comments and suggestions that make much improvement of the paper.

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Authors and Affiliations

  1. School of Mathematical Science, Shanxi University, Taiyuan, 030006, People’s Republic of China

    Shaoxing Sun & Xiaofei Qi

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  1. Shaoxing Sun
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  2. Xiaofei Qi
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Correspondence to Xiaofei Qi.

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Communicated by Palle Jorgensen.

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This work is partially supported by National Natural Science Foundation of China (12171290) and Fund Program for the Scientific Activities of Selected Returned Overseas Professionals in Shanxi Province (20200011).

This article is part of the topical collection “Harmonic Analysis and Operator Theory” edited by H. Turgay Kaptanoglu, Andreas Seeger, Franz Luef and Serap Oztop

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Sun, S., Qi, X. Higher Dimensional Numerical Range of \(\xi \)-Lie Products on Bound Self-adjoint Operators. Complex Anal. Oper. Theory 17, 20 (2023). https://doi.org/10.1007/s11785-022-01310-y

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  • DOI: https://doi.org/10.1007/s11785-022-01310-y

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