Crouzeix's conjecture

July 31 to August 4, 2017

at the

American Institute of Mathematics, San Jose, California

organized by

Mark Embree, Anne Greenbaum, and Michael Overton

Original Announcement

This workshop will be devoted to Crouzeix's conjecture and related topics concerning the numerical range and other $K$-spectral sets. Let $f$ denote any function analytic on $W(A)$, the numerical range of a square matrix $A$. Crouzeix's theorem, proved in 2007, states that
$$\|f(A)\| \le 11.08 \sup_{z\in W(A)} |f(z)|,$$ where $\|\cdot\|$ denotes the spectral norm. The constant $11.08$ is remarkable for being independent of the matrix dimension, thus allowing for easy passage to infinite dimensional settings. However, Crouzeix conjectured that this constant can be sharpened: $$\|f(A)\| \le 2 \sup_{z\in W(A)} |f(z)|.$$ This bound would generalize a result of Berger and Pearcy (1965) for integer powers, $f(z) = z^k$. Despite compelling numerical evidence, no proof has yet been found to reduce Crouzeix's constant below $11.08$.

The main topics for the workshop are:

Material from the workshop

A list of participants.

The workshop schedule.

Talk slides

Some Extensions of the Crouzeix-Palencia Result by Anne Greenbaum
Mapping theorems for numerical range by Thomas Ransford
The numerical range, Blaschke products and Compressions of the shift operator by Pamela Gorkin
Investigation of of Crouzeix’s Conjecture: Numerical Results and Variational Analysis by Michael Overton
Crouzeix’s conjecture holds for 3 x 3 matrices with elliptic numerical range centered at an eigenvalue by Christer Glader and Mikael Kurula
On admissible eigenvalue approximations from Krylov subspace methods for non-normal matrices by Jurjen Tebbens

Papers arising from the workshop:
Remarks on the Crouzeix-Palencia proof that the numerical range is a $(1+\sqrt{2})$-spectral set