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ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2020; 57(1): 69-79

Online first article October 21, 2019      Printed January 31, 2020

https://doi.org/10.4134/BKMS.b190054

Copyright © The Korean Mathematical Society.

On operators $T$ commuting with $CTC$ where $C$ is a conjugation

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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