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:


 * $$\|f(A)\| \le 2 \sup_{z\in W(A)} |f(z)|,$$

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


 * $$\|p(A)\| \le 2 \sup_{z\in W(A)} |p(z)|$$

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

History
Crouzeix's theorem, proved in 2007, states that:
 * $$\|f(A)\| \le 11.08 \sup_{z\in W(A)} |f(z)|$$

(the constant $$11.08$$ 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 $$1+\sqrt{2}$$, improving the original constant of $$11.08$$. The not yet proved conjecture states that the constant can be refined to $$2$$.

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.