Unitary similarity invariant function preservers of skew products of operators

Let B(H) denote the Banach algebra of all bounded linear operators on a complex Hilbert space H with dim⁡H≥3, and let A and B be subsets of B(H) which contain all rank one operators. Suppose F(⋅) is a unitary invariant norm, the pseudo spectra, the pseudo spectral radius, the C-numerical range, or the C-numerical radius for some finite rank operator C. The structure is determined for surjective maps Φ:A→B satisfying F(A⁎B)=F(Φ(A)⁎Φ(B)) for all A,B∈A. To establish the proofs, some general results are obtained for functions F:F1(H)∪{0}→[0,+∞), where F1(H) is the set of rank one operators in B(H), satisfying (a) F(μUAU⁎)=F(A) for a complex unit μ, A∈F1(H) and unitary U∈B(H), (b) for any rank one operator X∈F1(H) the map t↦F(tX) on [0,∞) is strictly increasing, and (c) the set {F(X):X∈F1(H) and ‖X‖=1} attains its maximum and minimum.