Open problem, Analysis, posed 1949

Invariant subspace problem

Open

Let $H$ be a complex separable Hilbert space of dimension $\dim H = \infty$ and let $T : H \to H$ be a bounded linear operator. Does there always exist a nontrivial closed linear subspace $W \subset H$ (meaning $W \neq \{0\}$ and $W \neq H$ ) that is invariant under $T$ , i.e., satisfies $T(W) \subseteq W$ ?

Posed by, 1949John von Neumann

Research frontier as of 2026

As of 2026, the invariant subspace problem for separable infinite-dimensional complex Hilbert spaces remains open. Positive results cover wide classes of operators—including normal operators, compact and polynomially compact operators, operators commuting with compact operators, subnormal operators, and contractions whose spectrum contains the unit circle $\mathbb{T}$ . Preprints posted in 2023 by Per Enflo (`arXiv:2305.15442`) and by Charles W. Neville (`arXiv:2307.08176`) claiming affirmative proofs for separable Hilbert spaces have not been confirmed by peer review.

Best known results

Lomonosov (1973): every nonscalar bounded operator on a Banach space that commutes with a nonzero compact operator has a nontrivial closed hyperinvariant subspace.
Scott Brown (1978) and Brown–Chevreau–Pearcy (1988): every subnormal operator on a Hilbert space, and every contraction $T$ whose spectrum $\sigma(T)$ contains the unit circle $\mathbb{T}$ , has a nontrivial closed invariant subspace.
Enflo (1976/1987) and Read (1985): there exist bounded linear operators on separable Banach spaces (including $\ell^1$ ) with no nontrivial closed invariant subspaces.

Tools and where they stop

Tool	Achieved	Where it stops
Schauder fixed-point theorem and compact approximation (Lomonosov technique)	Proves existence of hyperinvariant subspaces for operators commuting with nonzero compact operators	Many bounded operators on $H$ do not commute with any nonzero compact operator
Scott Brown's dual-algebra and $H^\infty$ functional calculus method	Settles the problem for subnormal operators and contractions with rich spectrum on $\mathbb{T}$	Requires a rich $H^\infty$ functional calculus or spectral set structure not available for general quasinilpotent operators

Open questions

Does every quasinilpotent bounded linear operator $T$ (meaning $\sigma(T) = \{0\}$ ) on a separable infinite-dimensional Hilbert space have a nontrivial closed invariant subspace?
Does every bounded linear operator on the reflexive sequence space $\ell^p$ for $1 < p < \infty$ , $p \neq 2$ , possess a nontrivial closed invariant subspace?

References

Victor I. Lomonosov (1973). Invariant subspaces for the family of operators which commute with a completely continuous operator · DOI:10.1007/BF01078890
Per Enflo (1987). On the invariant subspace problem for Banach spaces · DOI:10.1007/BF02392560
Heydar Radjavi, Peter Rosenthal (2003). Invariant Subspaces