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Isometries of Hilbert space valued function spaces

Part of: Normed linear spaces and Banach spaces; Banach lattices Linear function spaces and their duals

Published online by Cambridge University Press:  09 April 2009

Beata Randrianantoanina
Beata Randrianantoanina
Affiliation:
Department of Mathematics The University of Texas at AustinAustin, TX 78712 e-mail: [email protected]
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Abstract

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Let X be a (real or complex) rearrangement-invariant function space on Ω (where Ω = [0, 1] or Ω ⊆ N) 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: Ω → + K and σ: Ω → Ω and a strongly measurable operator map S of Ω into B (H) so that for almost all ω, S(ω) is a surjective isometry of H, and for any f ∈ X(H), T f(ω) = a(ω)S(ω)(f(σ(ω))) a.e. As a consequence we obtain a new proof of the characterization of surjective isometries in complex rearrangement-invariant function spaces.

MSC classification

Secondary: 46B04: Isometric theory of Banach spaces 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 61 , Issue 2 , October 1996 , pp. 150 - 161
Copyright
Copyright © Australian Mathematical Society 1996

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