On maps preserving the Jordan product of $C$-symmetric operators


  • Zouheir Amara Department of Mathematics, Labo LAGA, Faculty of Sciences, Mohammed First University, 60000 Oujda, Morocco
  • Mourad Oudghiri Department of Mathematics, Labo LAGA, Faculty of Sciences, Mohammed First University, 60000 Oujda, Morocco




Given a conjugation $C$ on a complex separable Hilbert space $H$, a bounded linear operator $A$ acting on $H$ is said to be $C$-symmetric if $A=CA^*C$. In this paper, we provide a complete description to all those maps on the algebra of linear operators acting on a finite dimensional Hilbert space that preserve the Jordan product of $C$-symmetric operators, in both directions, for every conjugation $C$ on $H$.


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Z. Amara and M. Oudghiri, Nonlinear preservers problem of complex symmetric operators, Asian-Eur. J. Math. 14 no. 9 (2021), Paper No. 2150162, 18 pp.  DOI  MR  Zbl

Z. Amara, M. Oudghiri, and K. Souilah, Complex symmetric operators and additive preservers problem, Adv. Oper. Theory 5 no. 1 (2020), 261–279.  DOI  MR  Zbl

A. Bourhim and M. Mabrouk, Maps preserving the local spectrum of Jordan product of matrices, Linear Algebra Appl. 484 (2015), 379–395.  DOI  MR  Zbl

A. Fošner, B. Kuzma, T. Kuzma, and N.-S. Sze, Maps preserving matrix pairs with zero Jordan product, Linear Multilinear Algebra 59 no. 5 (2011), 507–529.  DOI  MR  Zbl

S. R. Garcia, Conjugation and Clark operators, in Recent Advances in Operator-Related Function Theory, Contemp. Math. 393, Amer. Math. Soc., Providence, RI, 2006, pp. 67–111.  DOI  MR  Zbl

S. R. Garcia and M. Putinar, Complex symmetric operators and applications, Trans. Amer. Math. Soc. 358 no. 3 (2006), 1285–1315.  DOI  MR  Zbl

S. R. Garcia and J. E. Tener, Unitary equivalence of a matrix to its transpose, J. Operator Theory 68 no. 1 (2012), 179–203.  MR  Zbl

S. R. Garcia and W. R. Wogen, Complex symmetric partial isometries, J. Funct. Anal. 257 no. 4 (2009), 1251–1260.  DOI  MR  Zbl

Y. Ji, T. Liu, and S. Zhu, On linear maps preserving complex symmetry, J. Math. Anal. Appl. 468 no. 2 (2018), 1144–1163.  DOI  MR  Zbl

S. Jung, E. Ko, M. Lee, and J. Lee, On local spectral properties of complex symmetric operators, J. Math. Anal. Appl. 379 no. 1 (2011), 325–333.  DOI  MR  Zbl

B. Kuzma and T. Petek, Maps preserving unitarily invariant norms of Jordan product of matrices, J. Math. Anal. Appl. 455 no. 2 (2017), 1579–1596.  DOI  MR  Zbl

E. Prodan, S. R. Garcia, and M. Putinar, Norm estimates of complex symmetric operators applied to quantum systems, J. Phys. A 39 no. 2 (2006), 389–400.  DOI  MR  Zbl

X. Wang and Z. Gao, A note on Aluthge transforms of complex symmetric operators and applications, Integral Equations Operator Theory 65 no. 4 (2009), 573–580.  DOI  MR  Zbl

L. Zhao and J. Hou, Jordan zero-product preserving additive maps on operator algebras, J. Math. Anal. Appl. 314 no. 2 (2006), 689–700.  DOI  MR  Zbl