Crouzeix's conjecture

Template:Short description 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:

$\Vert f(A)\Vert \le 2\underset{z\in W(A)}{\sup }|f(z)|,$

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 $W(A)$. Slightly reformulated, the conjecture can also be stated as follows: for all square complex matrices $A$ and all complex polynomials $p$:

$\Vert p(A)\Vert \le 2\underset{z\in W(A)}{\sup }|p(z)|$

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

History

Crouzeix's theorem, proved in 2007, states that:

$\Vert f(A)\Vert \le 11.08\underset{z\in W(A)}{\sup }|f(z)|$

(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.

Further reading

• Template:Cite journal
• Template:Cite arXiv

See also

• Von Neumann's inequality