Rocky Mountain Journal of Mathematics
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.
Li, C. K., Poon, Y. T., & Sze, N. S. (2011). Elliptical range theorems for generalized numerical ranges of quadratic operators. The Rocky Mountain Journal of Mathematics, 813-832.