Bilinear isometries on spaces of vector-valued continuous functions

Let X, Y, Z be compact Hausdorff spaces and let E 1 , E 2 , E 3 be Banach spaces. If T : C ( X , E 1 ) × C ( Y , E 2 ) → C ( Z , E 3 ) is a bilinear isometry which is stable on constants and E 3 is strictly convex, then there exist a nonempty subset Z 0 of Z, a surjective continuous mapping h : Z 0 → X × Y and a continuous function ω : Z 0 → B i l ( E 1 × E 2 , E 3 ) such that T ( f , g ) ( z ) = ω ( z ) ( f ( π X ( h ( z ) ) ) , g ( π Y ( h ( z ) ) ) ) for all z ∈ Z 0 and every pair ( f , g ) ∈ C ( X , E 1 ) × C ( Y , E 2 ) . This result generalizes the main theorems in Cambern (1978) [2] and Moreno and Rodríguez (2005) [7].