Crouzeix's Conjecture

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

: $\, f(A)\, \le 11.08 \sup_ , f(z),$

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

: $\, f(A)\, \le 2 \sup_ , f(z),$

Michel Crouzeix and Cesar Palencia proved in 2017 that the result holds for $1+\sqrt$, improving the original constant of $11.08$.

Slightly reformulated, the conjecture can be stated as follows: For all square complex matrices $A$ and all complex polynomials $p$:

: $\, p(A)\, \le 2 \sup_ , p(z),$

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 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.

Further reading

*
*

References

{{Reflist

Conjectures
Matrix theory
Unsolved problems in mathematics