Elliptical Range Theorems for Generalized Numerical Ranges of Quadratic Operators

Chi-Kwong Li, William & Mary
Yiu-Tung Poon
Nung-Sing Sze

Abstract

The classical numerical range of a quadratic operator is an elliptical disk. This result is extended to different kinds of generalized numerical ranges. In particular, it is shown that for a given quadratic operator, the rank-k numerical range, the essential numerical range, and the q-numerical range are elliptical disks; the c-numerical range is a sum of elliptical disks, and the Davis-Wielandt shell is an ellipsoid with or without interior.