Into isometries of $C_0(X,E)$'s

Abstract

Suppose $X$ and $Y$ are locally compact Hausdorff spaces, $E$ and $F$ are Banach spaces and $F$ is strictly convex. We show that every linear isometry $T$ from $C_0(X,E)$ {\em into} $C_0(Y,F)$ is essentially a weighted composition operator $Tf(y)=h(y)(f(\phi (y)))$. This supplements results of Jerison (when $T$ is onto) and Cambern (when $X,Y$ are compact).