Some unitary similarity invariant sets preservers of skew Lie products

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