Crouzeix's Conjecture and Beyond for Special Classes of Matrices

Abstract

Let $A$ be an $n$ by $n$ matrix with numerical range $W(A) := \{ q^{*}Aq : q \in \mathbb{C}^n ,~\| q \|_2 = 1 \}$. We are interested in functions $\hat{f}$ that maximize $\| f(A) \|_2$ (the matrix norm induced by the vector 2-norm) over all functions $f$ that are analytic in $W(A)$ and satisfy $\max_{z \in W(A)} | f(z) | \leq 1$. It is known that there are functions $\hat{f}$ that achieve this maximum and that such functions are of the form $B \circ \phi$, where $\phi$ is any conformal mapping from $W(A)$ to the unit disk $\mathbb{D}$ and $B$ is a Blaschke product of degree at most $n-1$. Michel Crouzeix has conjectured that $\| \hat{f} (A) \|_2 \leq 2$ and he proved that $\| \hat{f} (A) \|_2 \leq 11.08$ [M. Crouzeix, Numerical range and functional calculus in Hilbert space, J. Funct. Anal., vol. 244, issue 2, 2007, p. 668-690]. Later Crouzeix and Palencia proved $\| \hat{f} (A) \|_2 \leq 1 + \sqrt{2}$ [M. Crouzeix and C. Palencia, The Numerical Range is a $(1+\sqrt{2})$-Spectral Set, SIMAX, vol. 38, 2017, p. 649-655]. However, the conjectured bound of $2$ remains unproven in general. Other questions about the optimal function $\hat{f}$ remain open as well. Here we consider questions about uniqueness of the optimizing function $\hat{f}$ and show that for $2$ by $2$ matrices, $\hat{f}$ is unique (up to multiplication by a scalar) but for certain $3$ by $3$ matrices this is not the case. It has been observed numerically that the optimal Blaschke product $\hat{B}$ associated with a given conformal mapping $\phi$ often has degree less than $n-1$, and we prove this for a certain class of matrices. We also evaluate $\| \hat{f} (A) \|_2$ explicitly for certain classes of matrices with elliptical numerical range. Our goal is to learn as much as possible about the optimal function $\hat{f}$, the associated Blaschke product $\hat{B}$, and the actual value of $\| \hat{f} (A) \|_2$ for matrices with elliptical numerical range. In doing this, we make use of known and new cyclic identities involving Jacobi elliptic functions.