A Lyapunov-like (linear) transformation L on a Euclidean Jordan algebra V is defined by the condition $$x \in K,\enspace y\in K^*,\quad \langle x,y\rangle =0\Rightarrow \langle L(x),y\rangle=0,$$where K is the symmetric cone of V. In this paper, we give an elementary proof (avoiding Lie algebraic ideas and results) of the fact that Lyapunov-like transformations on V are of the form La + D, where a ∈ V, D is a derivation, and La(x) = a ◦ x for all x ∈ V.
['Jiyuan Tao', 'M. Seetharama Gowda']
A REPRESENTATION THEOREM FOR LYAPUNOV-LIKE TRANSFORMATIONS ON EUCLIDEAN JORDAN ALGEBRAS