Crouzeix's conjecture

Crouzeix's conjecture is an unsolved problem in matrix analysis. It was proposed by Michel Crouzeix in 2004, and it can be stated as follows:
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. Slightly reformulated, the conjecture can also 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.

History

Crouzeix's theorem, proved in 2007, states that:
.
Michel Crouzeix and Cesar Palencia proved in 2017 that the result holds for, improving the original constant of. The not yet proved conjecture states that the constant can be refined to.

Special cases

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, 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.