Crouzeix's conjecture


Crouzeix's conjecture is an unsolved problem in matrix analysis. It was proposed by Michel Crouzeix in 2004, and it refines Crouzeix's theorem, which states:
where the set is the field of values of a n×n complex matrix and is a complex function, that is analytic in the interior of and continuous up to the boundary of. The constant is independent of the matrix dimension, thus transferable to infinite-dimensional settings. The not yet proved conjecture states that the constant is sharpable to :
Michel Crouzeix and Cesar Palencia proved in 2017 that the result holds for, improving the original constant of.
Slightly reformulated, the conjecture can be stated as follows: For all square complex matrices and all complex polynomials :
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 and for general n×n matrices that are nearly Jordan blocks. Furthermore, Anne Greenbaum and Michael L. Overton provided numerical support for Crouzeix's conjecture.