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

Let M-n,M-m be a rectangularfinite dimensional Cartan factor, i.e. the space L(C-n, C-m) with 1 <= n <= m, and let delta: M-n,M-m -> M-n,M-m be a quadratic Jordan derivation of M-n,M-m, i.e., a map (neither linearity nor continuity of delta is assumed) that satisfies the functional equation delta{ABA} = {delta(A) BA} + {A delta(B) A} + {AB delta(A)}, ( A, B is an element of M-n,M-m), where (A, B, C) -> {A B, C} := 1/2 (AB*C+ CB*A) stands for the Jordan triple product in M-n,M-m. We prove that then delta automatically is a continuous complex linear map on M-n,M-m. More precisely we show that delta admits a representation of the form delta(A) = UA + AV, (A is an element of M-n,M-m), for a suitable pair U, Vof square skew symmetric matrices with complex entries U. M-n,M-n and V is an element of M-m,M-m. (c) 2023 Elsevier Inc. All rights reserved.