Numerical ranges of Hankel matrices

In this paper, we study the numerical ranges of (finite) Hankel matrices and Hankel operators (on an infinite-dimensional space). The main concern is which nonempty bounded convex set △ in the plane is the numerical range W(A) of a Hankel matrix or a Hankel operator A. In Section 1 below, we prove results for △ a line segment, an elliptic disc, or a polygonal region. For example, we show that if △ is a closed elliptic disc in the plane, then a necessary and sufficient condition for the existence of an n-by-n Hankel matrix A_n with W(A_n) equal to △ for all n≥2 is that 0 is in △. In Section 2, we use the Megretskiĭ–Peller–Treil characterization of Hermitian Hankel operators to obtain an analogous condition for △ a (finite) line segment in the plane.