Conjecture Fieldbook

GlossaryStudy cards

Invariant subspace problem

Canonical problem dossier

Does every bounded linear operator on a complex separable infinite-dimensional Hilbert space have a non-trivial closed invariant subspace?

Status
Open
Field
Analysis
Verification
Reviewed dossier
Source grade
secondary
Reviewed
2026-07-27T18:04:33.834Z
Reviewer
Conjecture Fieldbook maintainers

Formal expression

{0}⊊M⊊H,T(M)⊆M\{0\}\subsetneq M\subsetneq H,\qquad T(M)\subseteq M

Why it matters

The problem probes whether linear operators in infinite dimensions must preserve some meaningful internal geometry.

Context and known approaches

Variants are solved or disproved on other Banach spaces, but the classical Hilbert-space formulation remains open.

Formal boundary

For every bounded T:H→H on a complex separable infinite-dimensional Hilbert space, does there exist closed M with {0}⊊M⊊H and T(M)⊆M?

Known partial results

Pending review. No structured partial-result note has been added to this dossier.

Source trail

  1. Wikipedia — Classical formulation and surrounding results.
  2. Wikipedia open-problems list — Composite index by mathematical field.
  3. Wikipedia list subsection — Placement in the accepted revision-pinned living-list snapshot.