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.

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