This workshop, sponsored by AIM
and the NSF, 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.
This event will be run as an AIM-style workshop. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.