analysis / Numerical range / Crouzeix's conjecture
Lorist and Schwenninger prove Crouzeix's conjecture (arXiv:2608.03841, Lemma 1) by combining a lower bound (their inequality (4)) with an upper bound (inequality (5)). In Remark 2 they observe that (5) alone gives $\kappa \le 1 + \sqrt{1 - \Re\langle E_1 Tx,x\rangle}$, so positivity of $\Re\langle E_1 Tx,x\rangle$ would prove the lemma outright. They write: "it is unclear whether $\Re\langle E_1 Tx,x\rangle \ge 0$ in general."
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Append-only history
Nothing in the paper's proof is affected. At the witness, inequality (4) holds with slack +27.0, inequality (5) holds with equality (f is inner), and the theorem itself holds with slack 2 - kappa = +0.80. Only the shortcut Remark 2 floats is refuted.
Research memory
Lorist and Schwenninger prove Crouzeix's conjecture (arXiv:2608.03841, Lemma 1) by combining a lower bound (their inequality (4)) with an upper bound (inequality (5)). In Remark 2 they observe that (5) alone gives $\kappa \le 1 + \sqrt{1 - \Re\langle E_1 Tx,x\rangle}$, so positivity of $\Re\langle E_1 Tx,x\rangle$ would prove the lemma outright. They write: "it is unclear whether $\Re\langle E_1 Tx,x\rangle \ge 0$ in general."
Nothing in the paper's proof is affected. At the witness, inequality (4) holds with slack +27.0, inequality (5) holds with equality (f is inner), and the theorem itself holds with slack 2 - kappa = +0.80. Only the shortcut Remark 2 floats is refuted.
Evidence graph
depends on · problem · claimed