The hyperinvariant subspace problem on Hilbert space

Lineage and corrections

Lean Comparator challenge BackwardIntertwiners (OpenAI math release) evidence for this event

The hyperinvariant subspace problem on Hilbert space parent of this event

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Theorem 1.1 and Corollary 1.2: on $L^2(\mathbb Z_2;\ell^2(\mathbb Z))$ a fibered bilateral weighted shift $S$ with quadratically decaying weights over the 2-adic odometer is nonzero with $\|S^n\|\le\exp(-10\lfloor(n+1)^2/4\rfloor)$, hence norm-quasinilpotent, and its commutant (containing a backward intertwiner $V$ and all scalar multipliers) is transitive. So every infinite-dimensional separable Hilbert space carries a nonzero quasinilpotent operator with no nonzero proper closed hyperinvariant subspace. The operator has many ordinary invariant subspaces (the fiberwise tails), so the invariant subspace problem itself is untouched. The companion reaches the same existence statement by a different construction. parent of this event

Backward intertwiners and a transitive commutant (companion) evidence for this event

Douglas-Pearcy 1972, Hyperinvariant subspaces and transitive algebras evidence for this event

VibeMathed record: The hyperinvariant subspace problem on Hilbert space evidence for this event

Zhu, Fang and Shi's question on invariant projections of weighted irrational rotations evidence for this event