... University, Garmsar, Iran2Laboratory of Theoretical Studies, Institute for Mathematical Research, University Putra Malaysia UPM,43400 Serdang, Selangor Darul Ehsan, Malaysia3Department of Mathematics, ... LabL a b, for all a, b A. Similarly, a linear mapping R : A Asatisfying RabaRb, for all a, b ∈Ais calledright centralizer on A. Adouble centralizer on A is a pair L, R, where L is a ... Banach algebra. Recall that a mappingL : A Ais a quadratic left centralizer if L is a quadratic homogeneous mapping, that is Lis quadratic and L a λ2L a , for all a ∈Aand λ ∈ C,andLabL a b2,...