Hyperinvariant subspace for absolutely norm attaining and absolutely minimum attaining operators

Neeru Bala, Golla Ramesh

Published 2020-02-21Version 1

A bounded linear operator $T:H_1\rightarrow H_2$, where $H_1,\,H_2$ are Hilbert spaces, is called \textit{norm attaining} if there exist $x\in H_1$ with unit norm such that $\|Tx\|=\|T\|$. If for every closed subspace $M\subseteq H_1$, the operator $T|_M:M\rightarrow H_2$ is norm attaining, then $T$ is called \textit{absolutely norm attaining}. If in the above definitions $\|T\|$ is replaced by the minimum modulus, $m(T):=\inf\{\|Tx\|:x\in H_1,\|x\|=1\}$, then $T$ is called \textit{minimum attaining} and \textit{absolutely minimum attaining}, respectively. In this article, we show the existence of a non-trivial hyperinvariant subspace for absolutely norm (minimum) attaining normaloid (minimaloid) operators.