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Two preprints claim a proof of Crouzeix’s conjecture

Two preprints give independent arguments for the sharp constant two in Crouzeix’s inequality: for a matrix or bounded operator A and an appropriate function f, the operator norm of f(A) is at most twice the maximum of |f| on A’s numerical range.

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Stable ID
math-crouzeix-conjecture-proof-2026
Revision
math-crouzeix-conjecture-proof-2026.v1
Field
Mathematics · Operator theory, functional analysis and numerical ranges
Evidence
Tier 1 · Peer reviewed: No
Record state
Provisional · Provisional claimed proof
Last checked

AI role

Lorist and Schwenninger disclose using ChatGPT 5.6 Pro to explore proof strategies; they wrote the note and accept responsibility. No AI disclosure was located on the inspected public page for Jin’s concurrent manuscript.

Record details

Problem or result
Crouzeix’s conjectured optimal constant two for functions of matrices and operators
Authors
Emiel Lorist and Felix Schwenninger; independently, Shanmu Jin
Institutions
Delft University of Technology and University of Twente; Jin’s affiliation was not verified from the inspected source
Result date
Lorist–Schwenninger preprint submitted August 4; Jin manuscript revised through August 7, 2026

Why it matters

The conjecture has been a central sharp-bound problem connecting non-normal matrices, spectral sets and numerical analysis. The factor two is known to be optimal, so a correct proof would close the remaining gap from the published 1 + square-root-of-two bound.

Limits

Both manuscripts are unreviewed. Lorist and Schwenninger’s arXiv posting is a five-page first version, and Jin’s posting is a Preprints.org manuscript rather than an independent specialist assessment. Concurrent arguments strengthen the signal but do not substitute for peer review.

Sources

  1. Primary: Lorist and Schwenninger, A solution to Crouzeix’s conjecture
  2. Independent preprint: Shanmu Jin, The Crouzeix Conjecture
  3. Published context: Crouzeix and Palencia, the earlier 1 + square-root-of-two bound

Correction and revision history

  1. 2026-08-08 — Added after primary-source review as a provisional claim; the scope, evidence level and material limitations are stated explicitly.

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