#
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.

### 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

Papers arising from the workshop: