On the Banach–Stone problem

Jyh-Shyang Jeang¹ ; Ngai-Ching Wong²

¹Institute of Mathematics Academia Sinica Taipei, Taiwan 11529 Republic of China
²Department of Applied Mathematics National Sun Yat-sen University Kaohsiung, Taiwan 80424 Republic of China

Studia Mathematica, Tome 155 (2003) no. 2, pp. 95-105

Résumé

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.

Détail

DOI : 10.4064/sm155-2-1

Mots-clés : locally compact hausdorff spaces banach spaces linear isometry provide three answers banach stone problem always written generalized weighted composition operator only strictly convex written weighted composition operator weak sense does contain copy ell infty written weighted composition operator classical sense