Preservers of radial unitary similarity functions on products of operators

Let S be a subset of the algebra L(H) of all bounded linear operators on an infinite-dimensional complex Hilbert space H, and let f:S→[0,∞) be a radial unitary similarity invariant function. Under some assumption on S and f, characterizations are obtained for surjective maps ϕ on S satisfyingf(ϕ(T)∘ϕ(S))=f(T∘S)(T,S∈S), where the binary operation ∘ stands for the product or the Jordan semi-triple product on operators. Analogous descriptions are obtained for the finite-dimensional case, without the surjectivity assumption on ϕ.