## 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 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.

Original language | English |
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Pages (from-to) | 60-74 |

Number of pages | 15 |

Journal | Linear Algebra and Its Applications |

Volume | 650 |

DOIs | |

State | Published - 1 Oct 2022 |

## Keywords

- Hankel matrix
- Hankel operator
- Numerical range