Abstract
A bounded linear operator \(T: {{\mathcal {H}}}\rightarrow {{\mathcal {H}}}\) is a C-normal operator if there exists a conjugation C on \({{\mathcal {H}}}\) such that \([CT, (CT)^{*}]=0\) where \([R,S]:=RS-SR.\) In this paper we study properties of C-normal operators. In particular, we prove that \(T-\lambda \) is C-normal for all \(\lambda \in {{\mathbb {C}}}\) if and only if T is a complex symmetric operator with the conjugation C. Moreover, we show that if T is C-normal, then the following statements are equivalent; (i) T is normal, (ii) T is quasinormal, (iii) T is hyponormal, (iv) T is p-hyponormal for \(0<p\le 1\). Finally, we consider operator transforms of C-normal operators.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Garcia, S.R.: Aluthge transforms of complex symmetric operators. Integr. Equ. Oper. Theory 60, 357–367 (2008)
Garcia, S.R.: Conjugation and Clark operators. Contemp. Math. 393, 67–112 (2006)
Garcia, S.R.: Means of unitaries, conjugations, and the Friedrichs operator. J. Math. Anal. Appl. 335, 941–947 (2007)
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)
Jung, I., Ko, E., Pearcy, C.: Aluthge transform of operators. Integr. Equ. Oper. Theory 37, 437–448 (2000)
Ko, E.: On operators with similar positive parts. J. Math. Anal. Appl. 463(1), 276–293 (2018)
McCarthy, C.A.: \({c_{p}}\). Isr. J. Math. 5, 249–271 (1967)
Ptak, M., Simik, K., Wicher, A.: \(C\)-normal operators. Electron. J. Linear Algebra 36, 67–79 (2020)
Wang, X., Gao, Z.: A note on Aluthge transforms of complex symmetric operators and applications. Integral Equ. Oper. Theory 65, 573–580 (2009)
Wang, C., Zhao, J., Zhu, S.: Remark on the structure of \(C\)-normal operators. Linear Multilinear Algebra (2020). https://doi.org/10.1080/03081087.2020.1771254
Acknowledgements
The authors would like to thank the referee for his/her valuable comments which helped to improve the paper. These authors contributed equally to this work. The first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government(MSIT) (2019R1F1A1058633) and the Ministry of Education (2019R1A6A1A11051177). The second author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(2019R1A2C1002653). The third author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2019R1A6A1A11051177) and (2020R1I1A1A01064575).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Thomas Schlumprech.
Rights and permissions
About this article
Cite this article
Ko, E., Lee, J.E. & Lee, MJ. On properties of C-normal operators. Banach J. Math. Anal. 15, 65 (2021). https://doi.org/10.1007/s43037-021-00147-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s43037-021-00147-5