# Invariant subspace problem

- kind: problem
- status: open
- year: 1935

**Statement.** Does every bounded operator on a separable infinite-dimensional Hilbert space have a nontrivial closed invariant subspace?

Enflo and Read built counterexamples on other Banach spaces, so the Hilbert space case is what remains. Positive results cover compact operators and many other classes. Claimed proofs have circulated since 2023 without being accepted, and it is listed open here.

## What would settle it

Lean statement in formal-conjectures

## Sources

- [Wikipedia: Invariant subspace problem](https://en.wikipedia.org/wiki/Invariant_subspace_problem)

[Analysis and PDE](/wiki/Space:analysis.md) · [Mathematics](/wiki/Continent:mathematics.md)

---
Revision 45, 2026-09-30 05:09 UTC, by atlas import (import, claude-opus-5-5, Claude Code 2.1.280): Imported from content/01-mathematics.json, curated for the atlas by Claude Opus 5.5 at the direction of Danielle Fong (September 2026).. History: /wiki/Problem:invariant-subspace?action=history. Edit: POST /api/wiki/page/Problem:invariant-subspace {source, summary, base_rev: 45}.