Nonlinear maps preserving the pseudo spectral radius of skew semi-triple products of operators

Let H be a complex Hilbert space of dimension greater than 2, and denote by L(H) the algebra of all bounded linear operators on H. For ε > 0 and T ∈ L(H), let rε(T) denote the ε-pseudo spectral radius of T. Let S1 and S2 be subsets of L(H) which contain all rank one operators and the identity. A characterization is obtained for surjective maps φ: S1 → S2 satisfying rε(φ(T)φ(S) ∗φ(T)) = rε(T S∗T) (T, S ∈ S1). An analogous description is also obtained for the pseudo spectrum of operators