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) the Banach algebra of all bounded linear operators on H. Let Delta(.) denote W(.) or sigma(epsilon)(.), where, for A, W(A) stands for the numerical range of A is an element of B(H) and sigma(epsilon)(A) the epsilon-pseudospectrum of A. It is shown that a bijective map (no algebraic structure assumed) Phi : B(H) -> B(K) satisfies that Delta(AB - BA*) = Delta(Phi(A)Phi(B) - Phi(B)Phi(A)*) for all A, B is an element of B(H) if and only if there exists a unitary operator U is an element of B(H, K) such that Phi(A) = mu U AU* for all A is an element of B(H), where mu is an element of {-1, 1}. If Delta(.) = W(.) then the injectivity assumption on Phi can be omitted. (C) 2014 Elsevier Inc. All rights reserved.