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:

$\|f(A)\|\leq 2\sup _{z\in W(A)}|f(z)|,$

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 $p$:

$\|p(A)\|\leq 2\sup _{z\in W(A)}|p(z)|$

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

History

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

$\|f(A)\|\leq 11.08\sup _{z\in W(A)}|f(z)|$

(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.

Further reading

• Ransford, Thomas; Schwenninger, Felix L. (2018-03-01). "Remarks on the Crouzeix–Palencia Proof that the Numerical Range is a $(1+{\sqrt {2}})$-Spectral Set". SIAM Journal on Matrix Analysis and Applications. 39 (1): 342–345. arXiv:1708.08633. doi:10.1137/17M1143757. S2CID 43945191.
• Gorkin, Pamela; Bickel, Kelly (2018-10-27). "Numerical Range and Compressions of the Shift". arXiv:1810.11680 [math.FA].

See also

• Von Neumann's inequality