Abstract. A bounded linear operator $T\colon H_1\rightarrow H_2$, where $H_1,H_2$ are Hilbert spaces, is said to be norm attaining if there exists a unit vector $x\in H_1$ such that $\|Tx\|=\|T\|$ and absolutely norm attaining (or $\mathcal {AN}$-operator) if $T|M\colon M\rightarrow H_2$ is norm attaining for every closed subspace $M$ of $H_1$. \par We prove a structure theorem for positive operators in $\beta (H):=\{T\in \mathcal B(H): T|_{M}\colon M\rightarrow M$ is norm attaining for all $M\in \mathcal R_{T}\}$, where $\mathcal R_T$ is the set of all reducing subspaces of~$T$. We also compare our results with those of absolutely norm attaining operators. Later, we characterize all operators in this new class.

DOI: 10.14232/actasm-020-426-9

AMS Subject Classification (1991): 47A15, 47A46; 47A10, 47A58

Keyword(s): compact operator, norm attaining operator, $\mathcal {AN}$-operator, reducing subspace

received 26.9.2020, revised 2.2.2021, accepted 5.2.2021. (Registered under 926/2020.)