Linear maps preserving $\mathcal{AN}$-operators
Bull. Korean Math. Soc. 2020 Vol. 57, No. 4, 831-838
Published online July 14, 2020
Printed July 31, 2020
Ramesh Golla, Hiroyuki Osaka
Sangareddy(Dist); Ritsumeikan University
Abstract : Let $H$ be a complex Hilbert space and $T : H\rightarrow H$ be a bounded linear operator. Then $T$ is said to be \textit{norm attaining} if there exists a unit vector $x_0\in H$ such that $\|Tx_0\|=\|T\|$. If for any closed subspace $M$ of $H$, the restriction $T|M : M \rightarrow H$ of $T$ to $M$ is norm attaining, then $T$ is called an \textit{absolutely norm attaining} operator or $\mathcal{AN}$-operator. In this note, we discuss linear maps on $\mathcal B(H)$, which preserve the class of absolutely norm attaining operators on $H$.
Keywords : Compact operator, isometry, $\mathcal{AN}$-operator, linear preserver problem
MSC numbers : Primary 15A86, 47B49, 47B07, 47B48; Secondary 47A10
Supported by : The second author was partially supported by KAKENHI Grant Number JP17K05285
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