Numerical Ranges of the product of Operators

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.