Math Conjectures

Invariant subspace problem for Hilbert space

OPENLandmarkOpen problemProposed 1935–1950 · Standard version

Canonical statement

For every separable infinite-dimensional complex Hilbert space HH and every bounded linear operator T:H→HT:H\to H, there is a closed linear subspace MM such that
{0}⊊M⊊H,T(M)⊆M. \{0\}\subsetneq M\subsetneq H,\qquad T(M)\subseteq M.
View source LaTeX
For every separable infinite-dimensional complex Hilbert space \(H\) and every bounded linear operator \(T:H\to H\), there is a closed linear subspace \(M\) such that \[ \{0\}\subsetneq M\subsetneq H,\qquad T(M)\subseteq M. \]
ANAL-FA-001:[Halmos1967InvariantSubspaces][ChalendarPartington2011ModernInvariant][ContinoGallardo2024HalmosProblem]
Analysis and functional analysis

Notes

This is the invariant subspace problem: does every bounded linear operator TT on a separable infinite-dimensional complex Hilbert space HH leave invariant some closed subspace MM other than {0}\{0\} and HH itself? The question took shape in the operator theory of the 1935–1950 period; no single first printed statement is securely identified, and it reached a wide audience through Halmos's problem book [Halmos1967InvariantSubspaces].

An affirmative answer is known for many classes of operators. Compact operators, normal operators (via the spectral theorem), polynomially compact operators, and numerous further Hilbert-space classes all possess nontrivial closed invariant subspaces; the main techniques are surveyed in [ChalendarPartington2011ModernInvariant]. In the opposite direction, the corresponding statement is false on several Banach spaces, where bounded operators without nontrivial closed invariant subspaces have been constructed, but those constructions have not been transferred to Hilbert space.

For an arbitrary bounded operator on Hilbert space the problem remains open; questions from Halmos's circle continue to attract attention [ContinoGallardo2024HalmosProblem], and recent manuscripts claiming a full solution have not been accepted.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.