Abstract
Given a conjugation C on a separable complex Hilbert space H, a bounded linear operator T on H is said to be C-symmetric if \(CTC=T^*\), and is said to be C-skew symmetric if \(CTC=-\,T^*\). In this paper, we provide a complete description of all additive maps, on the algebra of all bounded linear operators acting on H, that preserve C-symmetric operators for every conjugation C. We focus also on the linear maps preserving C-skew symmetric operators.
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References
Brešar, M.: Normal preserving linear mappings. Can. Math. Bull. 37, 306–309 (1994)
Cho, M., Lee, J.E., Motoyoshi, H.: On [m, C]-isometric operators. Filomat 31, 2073–2080 (2017)
Conway, J.B.: A Course in Functional Analysis. Springer, New York (1997)
Davi Berg, I.: An extension of the Weyl–von Neumann theorem to normal operators. Trans. Am. Math. Soc. 160, 365–371 (1971)
Garcia, S.R.: Conjugation and Clark operators. Contemp. Math. 393, 67–112 (2006)
Garcia, S.R., Putinar, M.: Complex symmetric operators and applications. Trans. Am. Math. Soc. 358, 1285–1315 (2006)
Garcia, S.R., Putinar, M.: Complex symmetric operators and applications II. Trans. Am. Math. Soc. 359, 3913–3931 (2007)
Garcia, S.R., Wogen, W.R.: Some new classes of complex symmetric operators. Trans. Am. Math. Soc. 362, 6065–6077 (2010)
Guterman, A., Li, C.-K., Šemrl, P.: Some general techniques on linear preserver problems. Linear Algebra Appl. 315, 61–81 (2000)
Ji, Y., Liu, T., Zhu, S.: On linear maps preserving complex symmetry. J. Math. Anal. Appl. 468, 1144–1163 (2018)
Jung, S., Ko, E., Lee, M., Lee, J.: On local spectral properties of complex symmetric operators. J. Math. Anal. Appl. 379, 325–333 (2011)
Li, C.G., Zhu, S.: Skew symmetric normal operators. Proc. Am. Math. Soc. 141, 2755–2762 (2013)
Mbekhta, M.: Survey on Linear (Additive) Preserver Problems. Advanced Courses of Mathematical Analysis IV, pp. 174–195. World Sci. Publ., Hackensack (2012)
Omladič, M., Šemrl, P.: Preserving diagonalisability. Linear Algebra Appl. 285, 165–179 (1998)
Oudghiri, M., Souilah, K.: Linear preservers of quadratic operators. Mediterr. J. Math. 13, 4929–4938 (2016)
Prodan, E., Garcia, S.R., Putinar, M.: Norm estimates of complex symmetric operators applied to quantum systems. J. Phys. A 39, 389–400 (2006)
Wang, X., Gao, Z.: A note on Aluthge transforms of complex symmetric operators and applications. Integr. Equ. Oper. Theory 65, 573–580 (2009)
Wark, H.M.: Spaces of diagonal operators. Math. Z. 237, 395–420 (2001)
Zagorodnyuk, S.M.: On a J-polar decomposition of a bounded operator and matrices of J-symmetric and J-skew-symmetric operators. Banach J. Math. Anal. 4, 11–36 (2010)
Zhu, S., Zhao, J.: The Riesz decomposition theorem for skew symmetric operators. J. Korean Math. Soc. 52, 403–416 (2015)
Acknowledgements
The authors would like to thank the referee for carefully reading our manuscript and making many valuable suggestions which served to improve this paper, especially for drawing our attention to the linear preservers problem of skew-symmetric operators.
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Communicated by Martin Mathieu.
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Amara, Z., Oudghiri, M. & Souilah, K. Complex symmetric operators and additive preservers problem. Adv. Oper. Theory 5, 261–279 (2020). https://doi.org/10.1007/s43036-019-00018-9
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DOI: https://doi.org/10.1007/s43036-019-00018-9