Description
A bounded linear operator $T: H_1\rightarrow H_2$, where $H_1,H_2$ are Hilbert spaces, is said to be norm attaining if there exists a unit vector $x\in H$ such that $\|Tx\|=\|T\|$ and absolutely norm attaining (or $\mathcal{AN}$-operator) if $T|M:M\rightarrow H_2$ is norm attaining for every closed subspace $M$ of $H$.
Let $\mathcal R_T$ denote the set of all reducing subspaces of $T$. Define
\begin{equation*}
\beta(H):={\{T\in \mathcal B(H): T|_{M}:M\rightarrow M \; is \; norm \; attaining \; \forall \; M\in \; \mathcal R_{T}}\}.
\end{equation*}
In this talk we introduce a structure theorem for positive operators in $\beta(H)$ and compare our results with those of absolutely norm attaining operators. Then, we characterize all operators in this new class. Lastly, we present the denseness of $\beta(H)$ in $B(H)$ and related topics.
Joint work with Golla Ramesh, IIT Hyderabad, India.