Every bounded operator on an infinite-dimensional Banach space has norm attainment

The problem asks whether there exists an infinite-dimensional Banach space $X$ such that every operator in $L(X)$ attains its norm. M. I. Ostrovskii posed this question; no proof or counterexample for the original statement is reported.

Known results

• James’s characterization implies that a hypothetical $X$ and $L(X)$ would both be reflexive.
• Holub proved that such an $X$ cannot have the approximation property; more broadly, it cannot have the bounded $\mathcal{A}$-approximation property for any nontrivial operator ideal $\mathcal{A}$.
• Kalton proved that $L(X)$ cannot be reflexive when $X$ is nonseparable.
• Thus possible examples must be separable reflexive spaces without suitable approximation properties.

September 2026 related preprint

A preprint claims that every bounded operator on $c_0$ and $\ell_1$ has a norm-attaining second adjoint, and distinguishes second- from higher-adjoint behavior on $c$. It does not resolve the original universal norm-attainment question, and the claim is unverified.

Current status (as of September 2026): The original existence problem remains open; only necessary obstructions and related second- and higher-adjoint results are recorded.