Separation Between the Ordinary and Strong Kreiss Constants

Question 6.1 of Chalmoukis, Tsikalas and Yakubovich asks how far the Power boundedness constant $P(T)$ of a matrix can exceed its ordinary Kreiss constant $K(T)$ . Answered more strongly: for every $K > 1$ there are matrices whose Cayley transforms satisfy $K(C_h(A_{n,h})) \le K$ while the strong Kreiss constant satisfies $K_s(C_h(A_{n,h})) \ge \tfrac{1}{2}Cn^{\alpha_K}$ with $\alpha_K = (K-1)/(C+K-1)$ . Since $P(T)\geq K_s(T)$ , this solves the question. Moreover, since the Kreiss matrix theorem gives $K_s(T) \le P(T) \le edK(T)$ in dimension $d$ , the exponent $\alpha < 1$ is optimal up to an arbitrarily small power loss.

Result: Proved
Status: Resolved
AI contribution: AI-assisted
Method: Construction
Field: Operator theory
Posed by: Nikolaos Chalmoukis, Georgios Tsikalas and Dmitry Yakubovich
Year posed: 2025
Years open: 1y
Solved: 2026-08-19
Model: ChatGPT 5.6 Pro, Claude Fable
Vendor: OpenAI
Collaborators: Emiel Lorist, Martin Meyries, Mark Veraar
Verification: Unreviewed
Publication: Preprint
Significance: 12 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What the AI did

The same disclosure as the inverse generator entry, since both fall to one paper: ChatGPT 5.6 Pro explored Schauder basis counterexamples, assisted the adaptation of Ansorena's work that produces the explicit basis in Proposition 2.1, and helped optimize the explicit constants. The formalization in Lean 4 was done using
Claude Fable by Anthropic. Note the boundary honestly - the disclosure names Proposition 2.1 and Theorem 1.1, not Theorem 1.4. Proposition 2.1 is the finite-dimensional construction every result in the paper is deduced from, including this one, so the model is in the loop for the machinery rather than for this theorem's derivation.

Verification

Checked by this site on 22 August 2026 against the v2 PDF (arXiv:2608.06272v2, 19 Aug): the paper states "We furthermore note that Theorem 1.4(i) solves [6, Question 6.1]" and gives the explicit constants quoted in the statement, and reference [6] is Chalmoukis, Tsikalas and Yakubovich, arXiv:2512.10025. The mathematics of Theorem 1.4 was not checked here, though a curator numerical check of Proposition 2.1 - the construction it is deduced from - was carried out for the sibling entry and confirmed its bounds up to n = 256.
A lean certificate of Theorem 1.1 was added in v3 of the paper (arXiv:2608.06272v2, 27 Aug)

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Same work resolves bothInverse generator problem

Separation Between the Ordinary and Strong Kreiss Constants

What the AI did

Verification

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