Crouzeix's conjecture

Crouzeix's conjecture is an unsolved (as of 2018)[1] problem in matrix analysis. It was proposed by Michel Crouzeix in 2004,[2] and it refines Crouzeix's theorem, which states:[3]

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

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 ${\displaystyle W(A)}$. The constant ${\displaystyle 11.08}$ is independent of the matrix dimension, thus transferable to infinite-dimensional settings. The not yet proved conjecture states that the constant is sharpable to ${\displaystyle 2}$:

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

Michel Crouzeix and Cesar Palencia proved in 2017 that the result holds for ${\displaystyle 1+{\sqrt {2}}}$,[4] improving the original constant of ${\displaystyle 11.08}$.

Slightly reformulated, the conjecture can 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.

While the general case is unknown, it is known that the conjecture holds for tridiagonal 3×3 matrices with elliptic field of values centered at an eigenvalue[5] 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.[7]