Let H and K be complex separable Hilbert spaces with dimensions at least three, and B(H) the Banach algebra of all bounded linear operators on H. Let Δ(·) denote W(?) or σ_ε(?), where, for A, W(A) stands for the numerical range of A ∈ B(H) and σ_ε(A) the e-pseudospectrum of A. It is shown that a bijective map (no algebraic structure assumed) Φ: B(H) → B(K) satisfies that △(AB - BA~*) = △(Φ(A）Φ(B） -Φ(B)Φ(A）~*） for all A, B ∈ B(H） if and only if there exists a unitary operator U ∈ B(H,K) such that Φ(A) = μUAU* for all A ∈ B(H), where μ ∈ {-1, 1}. If Δ(?) = W(?), then the injectivity assumption on Φ can be omitted.
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