Preservers Of Radial Unitary Similarity Functions On Products Of Self-adjoint Operators

In this paper we study the problem of characterizing the maps that preserve some radial unitary similarity function on Jordan semi-triple products of self-adjoint operators and the maps that preserve some radial unitary similarity function on Lie products of self-adjoint operators.Let H be a complex Hilbert space with dim H ≥ 3,Bs(H)be the Jordan algebra of all bounded self-adjoint operators on H.Let F:B(H)→[d,∞]with d ≥ 0 be a radial unitary similarity invariant function.In this paper we give a characterization of all surjective mapsφ:Bs(H)→Bs(H)that satisfy F(φ(A)φ(B)φ(A))= F(ABA)(?)A,B∈Bs(H),or satisfy F([φ(A),φ(B)])= F([A,B])(?)A,B∈Bs(H).Applying these two general results,we obtain a characterization of all surjective maps χ:Bs(H)→Bs(H)that preserve respectively the p-norm,the pseudo spectral radius and the pseudo spectrum on Jordan semi-triple products of self-adjoint operators,and obtain a characterization of all surjective maps φ:Bs(H)→Bs(H)that preserve respectively the p-norm and the pseudo spectral radius on Lie products of self-adjoint operators.