Abstract
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 σε(⋅), where, for A, W(A) stands for the numerical range of A∈B(H) and σε(A) the ε-pseudospectrum of A. It is shown that a bijective map (no algebraic structure assumed) Φ:B(H)→B(K) satisfies that Δ(AB−BA⁎)=Δ(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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