Isometric stability property of certain Banach spaces

1993/06/09 by Alexander Koldobsky, Koldobsky, Alexander
Mathematics · #46B #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:46B

paper · pdf · doi:10.48550/arxiv.math/9306208

arxiv created 1993/06/09 · arxiv updated 2016/09/06

Abstract

Let E be one of the spaces C(K) and L₁, F be an arbitrary Banach space, p>1, and (X,σ) be a space with a finite measure. We prove that E is isometric to a subspace of the Lebesgue-Bochner space L_p(X;F) only if E is isometric to a subspace of F. Moreover, every isometry T from E into L_p(X;F) has the form Te(x)=h(x)U(x)e, e∈ E, where h:X→ R is a measurable function and, for every x∈ X, U(x) is an isometry from E to F.

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