Math Conjectures

Crouzeix conjecture

OPENMajorExact constant problemProposed 2004 · Full conjecture

Canonical statement

For every complex Hilbert space HH, every T∈B(H)T\in B(H), and every complex polynomial pp,
∥p(T)∥≤2sup⁡z∈W(T)∣p(z)∣,W(T)={⟨Tx,x⟩:∥x∥=1}. \|p(T)\|\le 2\sup_{z\in W(T)}|p(z)|,\qquad W(T)=\{\langle Tx,x\rangle:\|x\|=1\}.
View source LaTeX
For every complex Hilbert space \(H\), every \(T\in B(H)\), and every complex polynomial \(p\), \[ \|p(T)\|\le 2\sup_{z\in W(T)}|p(z)|,\qquad W(T)=\{\langle Tx,x\rangle:\|x\|=1\}. \]
ANAL-FA-005:[Crouzeix2004AnalyticFunctionsMatrices][CrouzeixPalencia2017NumericalRange][RansfordSchwenninger2018Crouzeix]
Analysis and functional analysis

Notes

The Crouzeix conjecture, formulated by Michel Crouzeix in 2004, asserts that the numerical range W(T)W(T) of a Hilbert-space operator TT is a 22-spectral set: for every polynomial pp, the norm ∥p(T)∥\|p(T)\| is at most twice the supremum of ∣p∣|p| over W(T)W(T) [Crouzeix2004AnalyticFunctionsMatrices]. Since the closure of W(T)W(T) always contains the spectrum, this would give a clean functional-calculus bound governed only by the numerical range.

The constant 22 cannot be lowered: known examples show that no universal constant below 22 is possible, so the conjectured value would be sharp. In the other direction, Crouzeix and Palencia proved in 2017 that the numerical range is a (1+2)(1+\sqrt2)-spectral set, the best universal constant currently known [CrouzeixPalencia2017NumericalRange]; Ransford and Schwenninger subsequently revisited that proof [RansfordSchwenninger2018Crouzeix]. For normal operators the constant 11 is classical via the spectral theorem.

Closing the gap between 1+21+\sqrt2 and 22, for arbitrary operators and already for arbitrary matrices, remains open.

References (3)

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