Unitary similarity and maps commuting with the mean transform of triple product of operators

Let B(H) be the algebra of all bounded linear operators acting on a complex Hilbert space H, and ∼u be the unitary similarity relation on B(H). For any T∈B(H) with a polar decomposition T=VT|T|, the mean transform is M(T):=12(VT|T|+|T|VT). In this paper, we obtain the form of all bijective maps Φ on B(H) satisfying M(Φ(T)Φ(S)Φ(T))∼uΦ(M(TST)), (T,S∈B(H)). We further describe the form of all surjective maps Φ on B(H) satisfying M(Φ(T)Φ(S)Φ(T))∼uM(TST) for all T,S∈B(H).