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 ≥ 2 and Φ : B(H) → B(K) be a map whose range contains all operators of rank ≤ 1. It is shown that Φ satisfies that W ([Φ(A), Φ(B)]) = W ([A, B]) for any A, B ∈ B(H) if and only if dim H = dim K, there exist ε ∈ {1,−1}, a functional h : B(H) → C, a unitary operator U ∈ B(H, K), and a set S of operators in 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 Φ(A) =    εUAU∗ + h(A)I if A ∈ B(H) \ S, −εUAU∗ + h(A)I if A ∈ S, or Φ(A) =    iεUAtU∗ + h(A)I if A ∈ B(H) \ S, −iεUAtU∗ + h(A)I if A ∈ S, where A 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.