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 nÃÂn (i.e. square) 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.
Crouzeix's theorem, proved in 2007, states that:
(the constant 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 , improving the original constant of . More recently, dimension-dependent improvements have been obtained: Malman, Mashreghi, O'Loughlin and Ransford showed that for each fixed dimension there exists a constant such that the inequality holds for all matrices.
Related work connects the constant in Crouzeix-type inequalities to configuration constants arising from the NeumannâÂÂPoincaré operator and yields domain-dependent improvements of the CrouzeixâÂÂPalencia bound in certain settings. The not yet proved conjecture states that the constant can be refined to .
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 3ÃÂ3 matrices with elliptic field of values centered at an eigenvalue and for general nÃÂn matrices that are nearly Jordan blocks.. Furthermore, Anne Greenbaum and Michael L. Overton provided numerical support for Crouzeix's conjecture.