Claim: every bounded linear operator T on a complex, separable, infinite-dimensional Hilbert space H has a closed invariant subspace M (with {0} ≠ M ≠ H): T(M) ⊆ M.\n\nThe Banach-space version is false: Enflo (1975/87) and Read constructed operators without any invariant subspace.
Von Neumann handled normal operators in the 1930s; compact operators fell (Lomonosov 1973, in one of operator theory's most celebrated theorems). Enflo's 13-page May 2023 arXiv preprint claimed the Hilbert-space case in the affirmative, followed by Neville's independent claim two months later — but as a 2025 survey notes, neither has carried the field and the problem remains open at time of writing. The problem's stubbornness shaped decades of operator theory.
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