In minisymposium: Crouzeix's Conjecture
Mon 10:30–11:00, SW RaadzaalAbout my 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.
- M.Crouzeix, Numerical range and functional calculus in Hilbert space, Journal of Functional Analysis, vol. 244, 2007, pp. 668–690.
- M.Crouzeix, A functional calculus based on the numerical range. Applications, Linear and Multilinear Algebra, vol. 56, no. 1, 2008, pp. 81–103.