On the Uniqueness of Functions that Maximize the Crouzeix Ratio

Abstract

Let $A$ be an $n$ by $n$ matrix with numerical range $W(A):=\{q^{\ast }Aq:q\in {\mathbb{C}}^n,\Vert q{\Vert }_2=1\}$. We are interested in functions $\hat{f}$ that maximize $\Vert f(A){\Vert }_2$ (the matrix norm induced by the vector 2-norm) over all functions $f$ that are analytic in the interior of $W(A)$ and continuous on the boundary and satisfy ${\max }_{z\in W(A)}|f(z)|\le 1$. It is known that there are functions $\hat{f}$ that achieve this maximum and that such functions are of the form $B\circ \varphi$, where $\varphi$ is any conformal mapping from the interior of $W(A)$ to the unit disk $\mathbb{D}$, extended to be continuous on the boundary of $W(A)$, and $B$ is a Blaschke product of degree at most $n-1$. It is not known if a function $\hat{f}$ that achieves this maximum is unique, up to multiplication by a scalar of modulus one. We show that this is the case when $A$ is a $2\times 2$ nonnormal matrix or a Jordan block, but we give examples of some $3\times 3$ matrices with elliptic numerical range for which two different functions $\hat{f}$, involving the same conformal mapping but Blaschke products of different degrees, achieve the same maximal value of $||f(A)|{|}_2$.