Functional analysis · operator theory · Hilbert spaces

Invariant Subspace Problem for Complex Hilbert Space

Collaboration beta

The problem asks whether every bounded operator on an infinite-dimensional complex Hilbert space preserves some nonzero proper closed subspace. Many operator classes do, but no proof or counterexample is known in full generality.

{0} ⊊ M ⊊ H, T (M) \subseteq M

Known results and sources

The invariant subspace problem asks whether every bounded operator on infinite-dimensional complex Hilbert space preserves a nonzero proper closed subspace.

Research problem

Exact mathematical statement

Let $H$ be an infinite-dimensional complex Hilbert space and let $T\in\mathcal B(H)$ be a bounded linear operator. Must there exist a closed subspace $M$ such that

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

The subspace must be nonzero, proper, closed, and invariant under $T$ . Counterexamples on other Banach spaces and positive theorems for compact, normal, polynomially compact, or other special operator classes do not answer this exact Hilbert-space question.

Problem infographic

Problem at a glance

Problem-first functional-analysis diagram showing an arbitrary bounded operator, the sought nonzero proper closed invariant subspace, necessary counterexample constraints, known special classes, and two unresolved model branches. — Special operator classes have invariant subspaces, while the general complex Hilbert-space problem and the packet’s proper-support, full-support, and two-sided residual barriers remain open.

Current mathematical picture

Where work on Invariant Subspace Problem for Complex Hilbert Space stands

Open problem

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureSupport-insensitive logarithmic obstruction

The v5 audit notes that densities vanish off proper support, making global divergence automatic rather than contradictory. Local injectivity, invariant-slice avoidance, full-support elimination, and two-sided mixed-volume arguments remain viable in their exact branches.

Route status · Narrowed route

Main reductionConnected essential spectrum

Disconnected spectrum would produce a nontrivial Riesz projection, while Fredholm noninvertibility conflicts with injectivity and dense range.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeBreak operator-range avoidance in the proper-support one-sided branch.Task status · Ready to work on

Work mapped so far

Invariant Subspace Problem for Complex Hilbert Space in numbers

1.7kretained lines of mathematical investigation1,721 in the current working snapshot

Argument development: 1,453 · 84%
Explored or eliminated routes: 90 · 5%
Computational analysis: 3 · 0%
Open obligations: 59 · 3%
Definitions and setup: 116 · 7%

9selected mapped statements1routes investigated3open questions3contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Status checkedAug 14, 2026

Source recordedAug 14, 2026

Recommended next task

Break operator-range avoidance in the proper-support one-sided branch.

Suggested move: Use positivity, purity, tail comparison, infinite defect, or the backward-shift bridge to force a nonzero vector in Range Q^(1/2) supported on a proper positive-measure subset.

What would count as progress

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

13 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded

Narrowed routeSupport-insensitive logarithmic obstruction

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Break operator-range avoidance in the proper-support one-sided branch.Suggested move: Use positivity, purity, tail comparison, infinite defect, or the backward-shift bridge to force a nonzero vector in Range Q^(1/2) supported on a proper positive-measure subset.

Ready to work on

Close the logarithmic lower-tail contradiction in the full-support one-sided branch.Suggested move: Prove finite lower-tail integrability for one defect packet from the exact tail calculus and joint Bessel estimate, but only after full support is established.

Ready to work on

Find a genuinely two-sided obstruction in the C00 stable branch.Suggested move: Combine moving-boundary collapse for the operator and adjoint with a mixed Gram determinant or dual-volume lower bound; one-sided upper collapse alone is insufficient.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026

Current statusOpen problem

Whether every bounded operator on an infinite-dimensional complex Hilbert space has a nontrivial closed invariant subspace remains open. Many positive operator classes are known, and recent work adds sufficient criteria, but Banach-space counterexamples and special-class theorems do not settle the Hilbert-space problem.

[3][4]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2026
PreprintMüller and Tomilov proved a new invariant-subspace criterion when the spectrum contains the boundary of the numerical range, leaving the general problem open.[4]
2025
PreprintA recent survey reviewed the preceding fifteen years of progress while again stating that the Hilbert-space problem remains open.[3]
2011
Authoritative summaryChalendar and Partington’s monograph synthesized modern approaches and explicitly retained the complex Hilbert-space problem as open.[1]
1973
Peer reviewedLomonosov proved that a nonscalar bounded operator commuting with a nonzero compact operator has a nontrivial invariant subspace.[2]

Mathematical neighborhood

Related results and reusable starting points

Current focusInvariant Subspace Problem for Complex Hilbert Space

Solved special caseLomonosov compact-commutant theorem

Commutation with a nonzero compact operator guarantees a nontrivial invariant subspace but does not cover arbitrary bounded Hilbert-space operators.

[2]

Solved special caseNumerical-range boundary criterion

The numerical-range boundary condition gives a checkable sufficient criterion for several operator classes, not a universal theorem.

[4]

Related problemInvariant subspace problem on general Banach spaces

Banach-space counterexamples show that the analogous problem is false on some Banach spaces, but they do not answer the complex Hilbert-space question.

[3]

Formal and computational footholds

Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.

formal library support · partial resource linkedmathlib bounded-operator and Hilbert-space foundations
mathlib documents continuous linear maps, Hilbert spaces, adjoints, and spectral theory infrastructure, but the scoped search found no formal resolution of the general invariant subspace problem.
[5]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

Formalization targetA statement-aligned formalization of nontrivial closed T-invariant subspaces for arbitrary bounded operators on infinite-dimensional complex Hilbert spaces.
Formalization targetSubstantial infinite-dimensional operator theory beyond the available foundational continuous-linear-map and adjoint libraries.
Formalization targetA formally checked universal existence proof or a Hilbert-space counterexample; special-class formalizations would not suffice.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

7 standing statements2 proposed statements3 open questions1 narrowed routes

Statements by mathematical role9 selected mapped statements

theorem candidate1 of 91
reduction4 of 94
lemma2 of 92
negative result1 of 91
equivalence1 of 91

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.22 displayed rows · 1 route included

retained route statementMust every bounded complex Hilbert-space operator preserve a proper closed subspace?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementExact invariant-subspace questionintermediate
retained route statementEvery nonzero vector is cyclicintermediate
retained route statementNo eigenvalues and dense shifted rangesintermediate
retained route statementConnected essential spectrumintermediate
retained route statementOperator range avoids invariant slicesintermediate
retained route statementCanonical backward-shift bridgeintermediate
Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
Useful failureSupport-insensitive logarithmic obstructionreported failure
Research targetBreak operator-range avoidance in the proper-support one-sided branch.open
Research targetClose the logarithmic lower-tail contradiction in the full-support one-sided branch.open
Research targetFind a genuinely two-sided obstruction in the C00 stable branch.open
Narrowed routeSupport-insensitive logarithmic obstructionThe v5 audit notes that densities vanish off proper support, making global divergence automatic rather than contradictory. Local injectivity, invariant-slice avoidance, full-support elimination, and two-sided mixed-volume arguments remain viable in their exact branches.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeBreak operator-range avoidance in the proper-support one-sided branch.

Proper-support operator-range avoidance is the highest immediate audit target. The full-support lower-tail branch and the two-sided C00 branch remain independently open.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Continue the mathematics

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Prepared starting pointBreak operator-range avoidance in the proper-support one-sided branch.

Invariant Subspace Problem for Complex Hilbert Space · ready to start

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Current routes and known obstacles
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Sources and references5 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

1
Modern Approaches to the Invariant-Subspace Problemsurvey or monograph · Isabelle Chalendar, Jonathan R. Partington · Cambridge University Press · 2011 · DOI 10.1017/CBO9780511862434.001 · accessed Aug 14, 2026
2
Invariant subspaces for the family of operators which commute with a completely continuous operatorpeer reviewed result · V. I. Lomonosov · Functional Analysis and Its Applications · 1973 · DOI 10.1007/BF01080698 · MR MR0420305 · accessed Aug 14, 2026
3
Recent perspectives on the Invariant Subspace Problemsurvey or monograph · Isabelle Chalendar, Jonathan R. Partington · arXiv · 2025 · ARXIV 2507.21834 · accessed Aug 14, 2026
4
Invariant subspaces for operators with spectrum containing the boundary of the numerical rangepreprint · Vladimir Müller, Yuri Tomilov · arXiv · 2026-06-21 · ARXIV 2606.22715 · accessed Aug 14, 2026
5
Mathlib.Analysis.Normed.Operator.BoundedLinearMapsformalization · Lean mathematical library · accessed Aug 14, 2026

Important qualifications

The status review is scoped to bounded operators on infinite-dimensional complex Hilbert space; Banach-space counterexamples are treated only as neighboring context.
The 2026 numerical-range criterion is recorded as a preprint and a sufficient condition, not a general solution.
Scoped searches of current mathlib, Isabelle, and Coq public documentation found foundational operator theory but no end-to-end formalization or resolution of this problem; that does not establish absence.

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