Invariant projections, hyperinvariant subspaces, and transitive algebras

2 · Going further

A closed subspace is hyperinvariant for an operator $T$ if it is invariant under every bounded operator commuting with $T$. Manuscript 2 constructs, on every infinite-dimensional separable complex Hilbert space, a nonzero bounded operator $T$ with no nonzero proper closed hyperinvariant subspace. It is also norm-quasinilpotent: ${\lim }_{n\to \infty }\Vert T^n{\Vert }^{1/n}=0$. This decay condition does not say that $T$ itself is zero.

The commutant of $T$ is the collection of bounded operators commuting with it. Manuscript 2 states that this commutant is a proper, strongly closed, unital transitive complex operator algebra. Here transitivity expresses the absence of any nonzero proper closed subspace invariant under the whole algebra. The title identifies backward intertwiners as part of the approach, but the supplied abstract does not describe their construction or the steps of the argument.

Manuscript 1 starts instead with an arbitrarily prescribed irrational angle. It constructs a continuous nonnegative weight on the circle with exactly one zero and logarithmic integral $-\infty$. The manuscript shows that the corresponding weighted rotation is nonzero and norm-quasinilpotent and has no nontrivial invariant projection in the associated hyperfinite type ${\mathrm{II}}_1$ factor. Its abstract identifies this weight construction as the route to the counterexample, without providing further argument details.

The manuscripts therefore concern related but distinct settings: invariant projections inside a particular factor, and closed subspaces invariant under an operator's entire commutant. The supplied source marks Manuscript 2 as Lean-formalized; it does not establish formal verification of Manuscript 1 or explain a detailed deduction between the two constructions.

Useful notions

• Complex Hilbert spaces and closed subspaces
• Bounded operators and operator norms
• Commuting operators
• Projections and operator algebras
• Circle rotations and weighted operators

What was known before / what is new

Limits and assumptions

The Hilbert-space result assumes an infinite-dimensional separable complex Hilbert space and concerns hyperinvariant, not merely invariant, closed subspaces. The projection result concerns projections within the associated hyperfinite type II₁ factor and uses an irrational rotation with the stated weight conditions. Manuscript 2 is marked Lean-formalized, but the supplied material does not give a claim-by-claim inventory of that formalization; Manuscript 1 is not marked formalized. The abstracts provide construction ingredients rather than full arguments and do not specify further open questions.

Mini glossary

Hyperinvariant subspace

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A closed subspace preserved by every bounded operator commuting with a given operator.

Commutant

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The algebra of all bounded operators that commute with a given operator.

Norm-quasinilpotent

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An operator $T$ is norm-quasinilpotent when $\Vert T^n{\Vert }^{1/n}$ tends to zero as $n$ tends to infinity.

Invariant projection

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A projection whose range is preserved by the operator under consideration.

Transitive operator algebra

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An operator algebra with no nonzero proper closed subspace invariant under all its operators.

Keywords : Hyperinvariant subspaces · Invariant projections · Transitive commutants

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Manuscripts in this family