ILAS2016 — 11–15 July 2016 — KU Leuven, Belgium

## 20th ILAS Conference

In minisymposium: Crouzeix's Conjecture

Michel Crouzeix (Université de Rennes 1)

I have conjectured that the following inequality

\begin{equation*} \|p(A)\|\leq 2\,\max_{z\in W(A)}|p(z)| \end{equation*}

holds for all polynomials $p\in\mathbb C[z]$, for all $d\times d$ complex matrices $A$, and for all integer $d$. Here $\|M\|$ denotes the spectral norm of the matrix $M$ and $W(A)=\{v^*Av\,;v\in \mathbb C^d, v^*v=1\}$ is the numerical range of $A$. This inequality is well known if $p(z)=z$ (the spectral norm is bounded by twice the numerical radius); if $p(z)=z^n$, it follows from the power inequality (Berger and Pearcy). It has been proved in the $d=2$ case; but in the general case I have only succeeded to show [1] that there exists a best constant $\mathcal Q$ satisfying $2\leq\mathcal Q\leq 11.1$, such that, for all $p$, $A$, $d$,

\begin{equation*} \|p(A)\|\leq \mathcal Q\,\max_{z\in W(A)}|p(z)|. \end{equation*}

This estimate allows to construct a functional calculus which has applications in different domains of mathematics [2].

In this talk, I will describe different approaches and different mathematical tools used for getting bounds of this constant $\mathcal Q$. I will also provide some arguments supporting my conjecture and comment my numerical tests.

1. M.Crouzeix, Numerical range and functional calculus in Hilbert space, Journal of Functional Analysis, vol. 244, 2007, pp.  668–690.
2. M.Crouzeix, A functional calculus based on the numerical range. Applications, Linear and Multilinear Algebra, vol. 56, no. 1, 2008, pp. 81–103.