# 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:

• Theoretical approaches to improving Crouzeix's constant, including special cases and nonsmooth analysis.

• Careful numerical investigations of the conjecture using conformal maps and Blaschke products.

• Implications of the conjecture for matrix dilations, such as near-normal dilations of nonnormal matrices.

• Inverse problems for ensembles of points in the numerical range.

• Applications of these ideas to iterative methods in linear algebra.

## Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

A list of open problems.

### 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 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
Ritt operators, optimal constants and the inverse generator problem by Felix Schwenninger

Papers arising from the workshop:

The generalized numerical range of a set of matrices
by  Pan-Shun Lau, Chi-Kwong Li, Yiu-Tung Poon, and Nung-Sing Sze
Remarks on the Crouzeix-Palencia proof that the numerical range is a $(1+\sqrt2)$-spectral set
by  Thomas Ransford, and Felix Schwenninger,  SIAM J. Matrix. Anal. Appl. 39 (2018), 342–345  MR3769702
Spectral Sets: Numerical Range and Beyond
by  Michel Crouzeix, and Anne Greenbaum