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Analysis · Oct 6, 2026

The complete Crouzeix conjecture

Resolves the complete Crouzeix conjecture: for every bounded operator A on a complex Hilbert space and every finite matrix-valued polynomial P, one has ∥P[A]∥≤2sup⁡z∈W(A)∥P(z)∥\lVert P[A]\rVert\le2\sup_{z\in W(A)}\lVert P(z)\rVert, where W(A)W(A) is the numerical range. The constant 2 is sharp, independent of the matrix size, and valid in infinite dimensions.

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