Isometries between completely regular vector-valued function spaces

Abstract

In this paper, first we study surjective isometries (not necessarily linear) between completely regular subspaces A and B of C 0 (X, E) and C 0 (Y, F ) where X and Y are locally compact Hausdorff spaces and E and F are normed spaces, not assumed to be neither strictly convex nor complete. We show that for a class of normed spaces F satisfying a new defined property related to their T -sets, such an isometry is a (generalized) weighted composition operator up to a translation. Then we apply the result to study surjective isometries between A and B whenever A and B are equipped with certain norms rather than the supremum norm. Our results unify and generalize some recent results in this context.