Bilinear isometries on spaces of vector-valued continuous functions

Let X, Y, Z   be compact Hausdorff spaces and let E1E1, E2E2, E3E3 be Banach spaces. If T:C(X,E1)×C(Y,E2)→C(Z,E3)T:C(X,E1)×C(Y,E2)→C(Z,E3) is a bilinear isometry which is stable on constants and E3E3 is strictly convex, then there exist a nonempty subset Z0Z0 of Z  , a surjective continuous mapping h:Z0→X×Yh:Z0→X×Y and a continuous function ω:Z0→Bil(E1×E2,E3)ω:Z0→Bil(E1×E2,E3) such thatT(f,g)(z)=ω(z)(f(πX(h(z))),g(πY(h(z))))T(f,g)(z)=ω(z)(f(πX(h(z))),g(πY(h(z)))) for all z∈Z0z∈Z0 and every pair (f,g)∈C(X,E1)×C(Y,E2)(f,g)∈C(X,E1)×C(Y,E2). This result generalizes the main theorems in Cambern (1978) [2] and Moreno and Rodríguez (2005) [7].