The Inverse Generator Problem on Hilbert Spaces
If generates a bounded -semigroup on a Hilbert space and has dense range, does also generate a bounded -semigroup? Posed by deLaubenfels in 1988. Answered negatively: Lorist, Meyries and Veraar construct a bounded operator with dense range generating a bounded, strongly stable semigroup whose inverse generates no -semigroup at all. The counterexamples come from one explicit finite-dimensional construction, using bases of with uniformly bounded partial-sum projections but unconditionality constants growing like .
- Result
- Disproved(see note)
- Status
- Resolved
- AI contribution
- AI-assisted
- Method
- Construction
- Field
- Semigroup theory
- Posed by
- Ralph deLaubenfels
- Year posed
- 1988
- Years open
- 38y
- Solved
- 2026-08-06
- Model
- ChatGPT 5.6 Pro, Claude Fable 5
- Vendor
- OpenAI, Anthropic
- Collaborators
- Emiel Lorist, Martin Meyries, Mark Veraar
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 22 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
One finite-dimensional construction settles three related questions. Besides the inverse generator problem, it gives a generator whose Cayley transforms satisfy the ordinary Kreiss resolvent condition but are neither strongly Kreiss bounded nor power bounded, and it shows the Crank-Nicolson scheme is unstable in operator norm both over long times at fixed step size and under mesh refinement at fixed final time.
What the AI did
The paper's disclosure in full: "ChatGPT 5.6 Pro by OpenAI was used to explore proof strategies and to check intermediate steps. Claude Fable 5 by Anthropic was used to check the arguments and detect mistakes. The authors take full responsibility for the content of this note." Strategy exploration and checking rather than an attributed step, so the lower tier applies.
Verification
A preprint days old; nobody independent has checked it and there is no formalization. The construction is unusually checkable, though: every matrix entry has a closed formula, and a curator-run numerical check of Proposition 2.1 (the explicit finite-dimensional construction underlying Theorem 1.1) confirmed all of its claimed bounds - the Toeplitz inverse identity, uniform partial-sum projections, both multiplier bounds, and the n^alpha growth of the alternating multiplier - at sizes up to n = 256 for several alpha. That validates the engine of the counterexample, not the full operator-theoretic argument, so the entry stays unreviewed.
Source
Submitted by RustyKestrel290 on