Numerical Range of Lie Product of Operators

Denote by W(A) the numerical range of a bounded linear operator A, and \({[A, B] = AB - BA}\) the Lie product of two operators A and B. Let H,K be complex Hilbert spaces of dimension \({{\geq}2}\) and \({\Phi : {\mathcal{B}}(H) \to {\mathcal{B}}(K)}\) be a map whose range contains all operators of rank \({{\leq}1}\). It is shown that \({\Phi}\) satisfies that \({{W}([\Phi(A), \Phi(B)]) = {W}([A, B])}\) for any \({A, B \in {\mathcal{B}}(H)}\) if and only if dim H = dim K, there exist \({\varepsilon \in \{1, -1\}}\), a functional \({h : {\mathcal{B}}(H) \rightarrow {\mathbb{C}}}\), a unitary operator \({U \in {\mathcal{B}}(H, K)}\), and a set \({{\mathcal{S}}}\) of operators in \({{\mathcal{B}}(H)}\), that consists of operators of the form aP + bI for an orthogonal projection P on H if the dimension of H is at least 3, such that or where \({A^t}\) is the transpose of A with respect to an orthonormal basis of H. The proof of this result depends on the classifications of operators A or operator pairs A, B with some symmetric properties of W([A, B]) that are of independent interest.