Separation between the ordinary and strong Kreiss constants
A negative solution to the inverse generator problem on Hilbert spaces yields an operator whose Cayley transforms satisfy the ordinary Kreiss condition but are neither strongly Kreiss bounded nor power bounded.
- Lab
- Independent
- Model
- ChatGPT 5.6 Pro; Claude Fable
- Field
- Mathematics
- Date
- 2026-08-06
- Human collaborators
- Emiel Lorist, Martin Meyries, Mark Veraar
- Problem posed
- 2025 · open 1 yr
Sources
Original work
Independent commentary
What was found
The paper constructs a bounded operator with dense range on a Hilbert space that generates a bounded, strongly stable C0-semigroup while its inverse generates no C0-semigroup, which answers the inverse generator problem in the negative. It also builds an exponentially stable generator with 0 in the resolvent set whose inverse semigroup is unbounded and grows at least double logarithmically; for that generator every Cayley transform satisfies the ordinary Kreiss resolvent condition but is neither strongly Kreiss bounded nor power bounded, which separates the two notions. A numerical consequence follows: the Crank-Nicolson scheme is unstable in operator norm both for every fixed step size over long times and under mesh refinement at any fixed final time. All counterexamples are deduced from a common finite-dimensional construction.
Novelty check
The separation question between ordinary and strong Kreiss constants is attributed by vibemathed to Chalmoukis and Tsikalas and dated 2025, so the window for a prior answer is short. The inverse generator problem is the older and better-known target, and the paper answers it in the negative on Hilbert spaces, the setting where it was sharpest. Both claims descend from one explicit finite-dimensional construction whose entries are given by explicit formulas, so the construction can be inspected directly. No prior counterexample appears.
Caveats and known objections
A preprint, unrefereed, with no independent check on record; it has been revised, with v1 posted 2026-08-06 and v3 on 2026-08-26, and the date here is the first public posting. The formalization is partial and covers the finite-dimensional core rather than the claims of the entry: the proof of Theorem 1.1 has been formalized in Lean 4 in a public repository, while the semigroup and Kreiss deductions drawn from it are not machine-checked, so the grade stays at author-verified rather than formal. Autonomy is graded ai-assisted on a disclosure that keeps the model in a search and formalization role: ChatGPT 5.6 Pro was used to explore various Schauder basis counterexamples to the inverse generator problem, and the Lean 4 formalization was done using Claude Fable.
Also recorded at
vibemathedseparation-of-ordinary-and-strong-kreiss-constants
Nothing mechanically. It is a parallel listing of the same result, carrying its own verification label rather than an independent check.
Nobody outside the lab has checked this yet.
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Entry history (1 event)
- AddedEntered the registry graded Author verified and AI-assisted.
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Graded author verified for verification and ai-assisted for autonomy. What these mean.
Cite this entry
whataifound.org. (2026). Separation between the ordinary and strong Kreiss constants. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-08-06-kreiss-constant-separationBibTeX
@misc{whataifound-independent-2026-separation,
title = {Separation between the ordinary and strong Kreiss constants},
author = {{whataifound.org}},
year = {2026},
howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
note = {Result by Independent. Verification: Author verified. Autonomy: AI-assisted.},
url = {https://whataifound.org/finding/2026-08-06-kreiss-constant-separation}
}