Quasi-isometric embeddings of C 0 ( K , X ) $C_{0}(K, X)$ spaces which induce isometries whenever X $X$ is a Hilbert space

Suppose that $K$ and $S$ are locally compact Hausdorff spaces and $X$ is a Hilbert space. It is proven that if there exist real numbers $M\ge 1$, $L\ge 0$ and a map $T$ from $C_0(K,X)$ to $C_0(S,X)$ satisfying

In this case, as an immediate consequence, $\phi$ generates a linear isometry of $C_0\left(K\right)$ into $C_0\left(S_0\right)$. Even in the Lipschitz case ($L=0$), this result is the first nonlinear vector generalization of a classical Jarosz theorem (1984) concerning the into linear isomorphisms of spaces of continuous functions on locally compact Hausdorff spaces.