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 $\beta (H)$ of operators whose restrictions to all nonzero reducing subspaces attain their norms. We prove that $\beta (H)$ is dense in $\mathcal{B}(H)$ in the operator norm. Since $\beta (H)$ is not stable under arbitrary compact perturbations, we introduce a natural subclass ${\beta }_0(H)$ and investigate its perturbation properties. We show that ${\beta }_0(H)$ is stable under finite-rank perturbations and obtain a complete characterization of those compact perturbations that preserve membership in ${\beta }_0(H)$. We also determine the maximal compactly perturbation-invariant subset of ${\beta }_0(H)$. These results clarify the spectral structure and stability of norm-attaining operators on reducing subspaces.