On operators $T$ commuting with $CTC$ where $C$ is a conjugation
Bull. Korean Math. Soc. 2020 Vol. 57, No. 1, 69-79
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
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