Hyperinvariant subspace for absolutely norm attaining and absolutely minimum attaining operators

Abstract

A bounded linear operator $T$ defined on a Hilbert space $H$ is called \textit{norm attaining} if there exist $x\in H$ with unit norm such that $\|Tx\|=\|T\|$. If for every closed subspace $M\subseteq H$, the operator $T|_M:M\rightarrow H$ is norm attaining, then $T$ is called \textit{absolutely norm attaining}. If in the above definitions $\|T\|$ is replaced by the minimum modulus, $m(T):=\inf\{\|Tx\|:x\in H,\|x\|=1\}$, 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 attaining normaloid operators as well as absolutely minimum attaining minimaloid operators defined on a Hilbert space.Comment: We found a gap in Theorem 3.