Numerical ranges of products of two positive contractions

Let A and B be positive contractions on a Hilbert space. It was known before that the numerical range W(AB) of their product is contained in the rectangular region [−1/8,1]×[−1/4,1/4]. In this paper, we determine exactly when W(AB) touches on its four sides. This follows from the more general information on a containing region of W(AB) and on when they share a common tangent line. We also give an example of A and B on three-dimensional space for which W(AB) is not symmetric with respect to the x-axis.