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 equation δ{ABA}={δ(A)BA}+{Aδ(B)A}+{ABδ(A)}, (A,B ∈ Mn,m), where (A, B, C) →{A B, C} := 1/2 (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.