Numerical Ranges of the product of Operators

Hongke Du, Chi-Kwong Li, Kuo-Zhong Wang, Yueqing Wang, Ning Zuo

Published 2015-10-06Version 1

We study containment regions of the numerical range of the product of operators $A$ and $B$ such that $W(A)$ and $W(B)$ are line segments. It is shown that the containment region is equal to the convex hull of elliptical disks determined by the spectrum of $AB$, and conditions on $A$ and $B$ for the set equality holding are obtained. The results cover the case when $A$ and $B$ are self-adjoint operators extending the previous results on the numerical range of the product of two orthogonal projections.