On the derivations of the quadratic Jordan product in the space of rectangular matrices

Abstract

Let Mn,m be a rectangular finite dimensional Cartan factor, i.e. the space L(Cn,Cm) with 1≤n≤m, and let δ:Mn,m→Mn,m be a quadratic Jordan derivation of Mn,m, i.e., a map (neither linearity nor continuity of δ is assumed) that satisfies the functional equationδ{ABA}={δ(A)BA}+{Aδ(B)A}+{ABδ(A)},(A,B∈Mn,m), where (A,B,C)↦{AB,C}:=12(AB⁎C+CB⁎A) stands for the Jordan triple product in Mn,m. We prove that then δ automatically is a continuous complex linear map on Mn,m. More precisely we show that δ admits a representation of the form δ(A)=UA+AV, (A∈Mn,m), for a suitable pair U,V of square skew symmetric matrices with complex entries U∈Mn,n and V∈Mm,m.