N–fold Jordan product commuting maps with a λ –aluthge transform

Let B(H) be the set of all bounded linear operators from H to H, whereH is a complex Hilbert space. In this paper, we study the properties of T when the λ -Aluthge transform of Tn is T. Also we prove that the bijective map Φ:B(H)→B(K) commutes with a λ -Aluthge transform under the n-fold jordan product if and only if there exists an unitary operator U :H→K such that Φ(T)=UTU∗ for every T in B(H).