General Product Nonlinear Maps Commuting with the λ-Aluthge Transform

Let H be a complex Hilbert space and B( H) be the algebra of bounded linear operators on H. For n > 2 and T-1, T-2,..., T-n epsilon B( H), the operators are defined as follows: T1T2 ... Tn the usual product and T(1)0T(2)0 ... 0T(n) - 1/2 ( T1T2... T-n+ T-n... T2T1) the general Jordan product of T1,..., Tn. We give a complete characterization of the bijective maps Phi : B( H) -> B( K), where H, K are Hilbert spaces with dimH = 2, that satisfy..( F( T1) * F( T2) * u u u * F( Tn)) = F(..( T1 * T2 * u u u * Tn)) for all T1, T2,..., Tn. B( H), where..( T) is the.- Aluthge transform of T. B( H) and T1 * u u u * Tn stands for the usual product or the general Jordan product of T1,..., Tn. We show that there exists a unitary operator U : H. K and a constant a with a n- 1 = 1, such that F has the form F( T) = aUTU* for all T. B( H).