{"id":"openai-math/The-complete-Crouzeix-theorem","pdf":"https://github.com/openai/math/blob/main/preprints/The-complete-Crouzeix-theorem-September-23-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/The-complete-Crouzeix-theorem-September-23-2026","tags":["numerical-analysis","operator-theory","numerical-range"],"score":0,"title":"The complete Crouzeix theorem: optimal similarity and a common positive boundary representation","domain":"Functional analysis","family":"325","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.FA","summary":"We resolve the complete Crouzeix conjecture by proving the sharp constant-two numerical-range inequality for every bounded operator on a complex Hilbert space and every matrix-valued polynomial. No separability assumption is needed. The closure of the numerical range is a complete 2-spectral set, and the bound extends to finite matrix-valued functions holomorphic near that closure. For a finite matrix and a bounded convex domain containing its numerical range with regular real-analytic Jordan boundary, the optimal similarity making its conformal disk image contractive is attained with…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-23","reactions":{},"formalized":true,"family_title":"The complete Crouzeix conjecture"}