Crouzeix's conjecture

Crouzeix's conjecture is an unsolved problem in matrix analysis. It was proposed by Michel Crouzeix in 2004,^[1] and it can be stated as follows:

${\displaystyle \|f(A)\|\leq 2\sup _{z\in W(A)}|f(z)|,}$

where the set ${\displaystyle W(A)}$ is the field of values of a n×n (i.e. square) complex matrix ${\displaystyle A}$ and ${\displaystyle f}$ is a complex function that is analytic in the interior of ${\displaystyle W(A)}$ and continuous up to the boundary of ${\displaystyle W(A)}$. Slightly reformulated, the conjecture can also be stated as follows: for all square complex matrices ${\displaystyle A}$ and all complex polynomials ${\displaystyle p}$:

${\displaystyle \|p(A)\|\leq 2\sup _{z\in W(A)}|p(z)|}$

holds, where the norm on the left-hand side is the spectral operator 2-norm.

Crouzeix's theorem, proved in 2007, states that:^[2]

${\displaystyle \|f(A)\|\leq 11.08\sup _{z\in W(A)}|f(z)|}$

(the constant ${\displaystyle 11.08}$ is independent of the matrix dimension, thus transferable to infinite-dimensional settings).

Michel Crouzeix and Cesar Palencia proved in 2017 that the result holds for ${\displaystyle 1+{\sqrt {2}}}$,^[3] improving the original constant of ${\displaystyle 11.08}$. More recently, dimension-dependent improvements have been obtained: Malman, Mashreghi, O'Loughlin and Ransford showed that for each fixed dimension ${\displaystyle N}$ there exists a constant ${\displaystyle C_{N}<1+{\sqrt {2}}}$ such that the inequality holds for all ${\displaystyle N\times N}$ matrices.^[4]

Related work connects the constant in Crouzeix-type inequalities to configuration constants arising from the Neumann–Poincaré operator and yields domain-dependent improvements of the Crouzeix–Palencia bound in certain settings.^[5]

The not yet proved conjecture states that the constant can be refined to ${\displaystyle 2}$.

While the general case is unknown, it is known that the conjecture holds for some special cases. For instance, it holds for all normal matrices,^[6] for tridiagonal 3×3 matrices with elliptic field of values centered at an eigenvalue^[7] and for general n×n matrices that are nearly Jordan blocks.^[6]. Furthermore, Anne Greenbaum and Michael L. Overton provided numerical support for Crouzeix's conjecture.^[8]

• Ransford, Thomas; Schwenninger, Felix L. (2018-03-01). "Remarks on the Crouzeix–Palencia Proof that the Numerical Range is a ${\displaystyle (1+{\sqrt {2}})}$-Spectral Set". SIAM Journal on Matrix Analysis and Applications. 39 (1): 342–345. arXiv:1708.08633. doi:10.1137/17M1143757. S2CID 43945191.
• Gorkin, Pamela; Bickel, Kelly (2018-10-27). "Numerical Range and Compressions of the Shift". arXiv:1810.11680 [math.FA].

• Von Neumann's inequality