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⁎Aq:q∈Cn,‖q‖2=1}. We are interested in functions fˆ that maximize ‖f(A)‖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 maxz∈W(A)⁡|f(z)|≤1. It is known that there are functions fˆ that achieve this maximum and that such functions are of the form B∘ϕ, where ϕ is any conformal mapping from the interior of W(A) to the unit disk 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 fˆ 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 2×2 nonnormal matrix or a Jordan block, but we give examples of some 3×3 matrices with elliptic numerical range for which two different functions fˆ, involving the same conformal mapping but Blaschke products of different degrees, achieve the same maximal value of ||f(A)||2.