Preservers of pseudo spectral radius of operator products

Let HH be an infinite-dimensional complex Hilbert space and let L(H)L(H) be the algebra of all bounded linear operators on HH. For ε>0ε>0 and T∈L(H)T∈L(H), let rε(T)rε(T) denote the ε-pseudo spectral radius of T. We characterize surjective maps ϕ   on L(H)L(H) which satisfyrε(ϕ(T)ϕ(S))=rε(TS)rε(ϕ(T)ϕ(S))=rε(TS) for all T,S∈L(H)T,S∈L(H). We also obtain analogous result for the finite-dimensional case, without the surjectivity assumption on ϕ.