Isometries of Hilbert space valued function spaces

1994/11/16 by Beata Randrianantoanina, Randrianantoanina, Beata
Mathematics · #46B #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.math/9411210

openalex publication_date 1994/11/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X be a (real or complex) rearrangement-in\-va\-riant function space on \Om (where \Om = [0,1] or \Om ⊆ \bbN) whose norm is not proportional to the L₂-norm. Let H be a separable Hilbert space. We characterize surjective isometries of X(H). We prove that if T is such an isometry then there exist Borel maps a:\Om→\bbK and σ:\Om\lra\Om and a strongly measurable operator map S of \Om into \calB(H) so that for almost all \om S(\om) is a surjective isometry of H and for any f∈ X(H) Tf(\om)=a(\om)S(\om)(f(σ(\om))) a.e. As a consequence we obtain a new proof of characterization of surjective isometries in complex rearrangement-invariant function spaces.

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