## On the Banach–Stone problem

### Volume 155 / 2003

Studia Mathematica 155 (2003), 95-105
MSC: 46B04, 46E40, 46E15.
DOI: 10.4064/sm155-2-1

#### Abstract

Let $X$ and $Y$ be locally compact Hausdorff spaces, let $E$ and $F$ be Banach spaces, and let $T$ be a linear isometry from $C_0(X,E)$
*into* $C_0(Y,F)$. We provide three new answers to the Banach–Stone problem: (1) $T$ can always be written as a generalized weighted composition operator if and only if $F$ is strictly convex; (2) if $T$ is onto then $T$ can be written as a weighted composition operator in a {weak} sense; and (3) if $T$ is onto and $F$ does not contain a copy of $\ell _2^\infty $ then $T$ can be written as a weighted composition operator in the classical sense.