Abstract

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 An with W(An) 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.

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.