The Stability and Density of Norm Attaining Operators on Reducing Subspaces

For a bounded linear operator on an infinite-dimensional Hilbert space, we consider the class $β(H)$ of operators whose restrictions to all nonzero reducing subspaces attain their norms. We prove that $β(H)$ is dense in $\mathcal{B}(H)$ in the operator norm. Since $β(H)$ is not stable under arbitrar