On operators $T$ commuting with $CTC$ where $C$ is a conjugation
Bull. Korean Math. Soc. 2020 Vol. 57, No. 1, 69-79
https://doi.org/10.4134/BKMS.b190054
Published online January 31, 2020
Muneo Ch\=o, Eungil Ko, Ji Eun Lee
Kanagawa University; Ewha Womans University; Sejong University
Abstract : In this paper, we study the properties of $T$ satisfying $[CTC$, $T]=0$ for some conjugation $C$ where $[R,S]:=RS-SR.$ In particular, we show that if $T$ is normal, then $[CTC,C]=0$. Moreover, the class of operators $T$ satisfy $[CTC,T]=0$ is norm closed. Finally, we prove that if $T$ is complex symmetric, then $T$ is binormal if and only if $[C|T|C,|T|]=0$.
Keywords : Conjugation operator, complex symmetric operator, normal operator
MSC numbers : Primary 47A05; Secondary 47B20
Supported by : This work is supported by Grant-in-Aid Scientific Research No.15K04910. The second author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(2016R1D1A1B03931937).
The third 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).
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