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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

References

[1]Calvert, B. and Fitzpatrick, S., ‘Characterizing l p and c 0 by projections onto hyperplanes’, Boll. Un. Math. Ital. 5 (1986), 405–410.Google Scholar
[2]Cambern, M., ‘The isometries of L p (X; K)’, Pacific J. Math. 55 (1974), 9–17.CrossRefGoogle Scholar
[3]Cambern, M., ‘Isometries of measurable functions’, Bull. Austral. Math. Soc. 24 (1981), 13–26.Google Scholar
[4]Fleming, R. J. and Jamison, J. E., ‘Isometries of Banach spaces – a survey’, in: Analysis, geometry and groups: a Riemann legacy (Hadronic Press, Palm Harbor, 1993) pp. 52–123.Google Scholar
[5]Fleming, R. J., ‘Classes of operators on vector-valued integration spaces’, J. Austral. Math. Soc. 24 (1977), 129–138.Google Scholar
[6]Goldstein, J., ‘Groups of isometries on Orlicz spaces’, Pacific J. Math. 48 (1973), 387–393.CrossRefGoogle Scholar
[7]Greim, P., ‘Isometries and L p-structure of separably valued Bochner L p-spaces’, in: Measure theory and its applications. Proc. Conf. Sherbrooke 1982 (eds. Belley, J. M., Dubois, J. and Morales, P.), Lecture Notes in Math. 1033 (Springer, Berlin, 1983) pp. 209–218.Google Scholar
[8]Jamison, J. E. and Loomis, I., ‘Isometries of Orlicz spaces of vector valued functions’, Math. Z. 73 (1986), 33–39.Google Scholar
[9]Kalton, N. J. and Randrianantoanina, B., ‘Surjective isometries of rearrangement-invariant spaces’, Quart. J. Math. Oxford Ser. (2) 45 (1994), 301–327.CrossRefGoogle Scholar
[10]Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces, vol. 2: Function spaces (Springer, Berlin, 1979).Google Scholar
[11]von Neumann, J., ‘Einige Sätze über messbare Abbildungen’, Ann. of Math. 33 (1932), 574–586.Google Scholar
[12]Randrianantoanina, B., Isometries of function spaces (Ph.D. Thesis, University of Missouri-Columbia, 1993).Google Scholar
[13]Rosenthal, H. P., ‘Contractively complemented subspaces of Banach spaces with reverse monotone (transfinite) bases’, Longhorn Notes, The University of Texas Functional Analysis Seminar (1984–1985), 1–14.Google Scholar
[14]Rosenthal, H. P., ‘Functional Hilbertian sums’, Pasific J. Math. 124 (1986), 417–467.CrossRefGoogle Scholar
[15]Sourour, A. R., ‘The isometries of L p(ω, X)’, J. Funct. Anal. 30 (1978), 276–285.CrossRefGoogle Scholar
[16]Tam, K. W., ‘Isometries of certain function spaces’, Pacific J. Math. 31 (1969), 233–246.CrossRefGoogle Scholar
[17]Zaidenberg, M. G., ‘On the isometric classification of symmetric spaces’, Soviet Math. Dokl. 18 (1977), 636–640.Google Scholar
[18]Zaidenberg, M. G., Special representations of isometries of function spaces (in Russian) Studies in the theory of functions of several variables No. 174 (Yaroslavl, 1980) pp. 84–91.Google Scholar