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OP-10446Status unassessedAnalysis

Investigation of the Crouzeix-Palencia proof · Question 1.7

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

In what situations (other than in Caldwell-Greenbaum-Li) does the estimate

∥f(A)+g(A)∗∥≥∥f(A)∥\|f(A)+g(A)^{*}\|\geq \|f(A)\|

hold? Here, f=B∘ϕf=B\circ \phi is such that BB is a Blaschke product, ϕ\phi is a conformal mapping from Ω\Omega to D\mathbb{D} and such that ∥f(A)∥\|f(A)\| is maximized over all all ∥f∥Ω=1\|f\|_{\Omega}=1.

Lean 4

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Background

Section context

This section deals with the careful analysis of the proof in order to think of possible adaption in order to improve the constant.

Progress & source assessment

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

Status note in the source

In this case, the conjecture holds.

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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Discussion

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