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OP-10458Status unassessedAnalysis

Optimal constant dependence for Ritt-operators · Question 8.1

Crouzeix's conjecture · AIMPL workshop record

MSC 47 · Operator theory

Editorial mapping of AIMPL list topic: Crouzeix's conjecture. http://aimpl.org/crouzeix/ Not an author-supplied MSC code.

Workshop record · status unassessedAs of 2026-09-10 source snapshot; not a current-status certificationSource & evidence ↓
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Problem statement

Is this CC-dependence optimal? What about finite dimensions? In the examples I know the dependence is linear.

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Background

Section context

A problem that is a bit less connected to Crouzeix's conjecture, but may require similar techniques coming from function theory.

Question context

Let TT be an operator on a Banach space XX and assume that the spectrum is contained in Dˉ\bar{\mathbb{D}} as well as C=sup⁡∣z∣>1∥(z−1)(zI−T)−1∥<∞C=\sup_{|z|>1}\|(z-1)(zI-T)^{-1}\|<\infty. Then, using contour integrals, it can be shown that

sup⁡n∈N∥Tn∥≤KClog⁡(1+C)\sup_{n\in\mathbb{N}}\|T^{n}\|\leq K C \log(1+C)

for some absolute constant KK, see Bakaev'03 ("Constant-size control in stability estimates under some resolvent estimates" in "Numerical methods and programming", 2003 (4)), and Source citation: MR3474352).

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Dated source record, inspected 2026-09-10. Current status has not been independently reassessed. Source-reported results have not been independently verified.

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References
  1. [1]Crouzeix's conjecture · Original question link
  2. [2]AIMPL source bibliography link
  3. [3]AIMPL text license · CC BY-SA 3.0 linkAIM Problem Lists (AIMPL), original contributors; CC BY-SA 3.0. Adaptation: structural TeX converted to Markdown, documented source macros expanded, and source bibliography links added. Mathematical wording retained. These adapted source texts remain under CC BY-SA 3.0.

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