Jordan triple homomorphisms on $${\mathcal {T}}_\infty (F)$$

Abstract Let $${\mathcal {T}}_\infty (F)$$ T∞(F) be the algebra of all $${\mathbb {N}}\times {\mathbb {N}}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">N×N upper triangular matrices defined over a field F characteristic different from 2. We consider Jordan triple homomorphisms , i.e. additive maps that satisfy condition $$\phi (xyx)=\phi (x)\phi (y)\phi (x)$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">ϕ(xyx)=ϕ(x)ϕ(y)ϕ(x) for $$x,y\in {\mathcal xmlns:mml="http://www.w3.org/1998/Math/MathML">x,y∈T∞(F) . For case when is prime we find form such $$ xmlns:mml="http://www.w3.org/1998/Math/MathML">ϕ general present surjective