Maps preserving the maximal numerical range of generalized products of operators

Let H be a complex Hilbert space and B(H) be the algebra of all bounded linear operators on it. T1∗T2∗⋯∗Tk denotes the generalized product of k operators T1,T2,…,Tk∈B(H). For an operator T∈B(H), Ttr stands for the transpose with respect to a fixed but arbitrary orthonormal basis of H. Denote by W0(T) the maximal numerical range of T. We characterize maps φ on B(H) whose ranges contain every operator of rank no greater than two and which satisfy W0(φ(T1)∗φ(T2)∗⋯∗φ(Tk))=W0(T1∗T2∗⋯∗Tk) for all T1,T2,…,Tk∈B(H). We show that such a map φ must be of the form T↦αUTU∗ or T↦αUTtrU∗, where U∈B(H) is a unitary operator and α∈C is a scalar with αm=1. However, T1∗T2∗⋯∗Tk is a generalized semi Jordan product in the latter case. When the generalized product of operators is replaced with the skew-generalized product of operators, we reach the same conclusion but with αm−1α¯=1.