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Extension of isometries on the unit sphere of L p spaces

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Abstract

In this paper we study the isometric extension problem and show that every surjective isometry between the unit spheres of L p(µ) (1 < p < ∞, p ≠ 2) and a Banach space E can be extended to a linear isometry from L p(µ) onto E, which means that if the unit sphere of E is (metrically) isometric to the unit sphere of L p(µ), then E is linearly isometric to L p(µ). We also prove that every surjective 1-Lipschitz or anti-1-Lipschitz map between the unit spheres of L p(µ1, H 1) and L p(µ2, H 2) must be an isometry and can be extended to a linear isometry from L p(µ1, H 1) to L p(µ2, H 2), where H 1 and H 2 are Hilbert spaces.

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Authors and Affiliations

  1. Department of Mathematics, Tianjin University of Technology, Tianjin, 300384, P. R. China

    Dong Ni Tan

  2. School of Mathematical Science, Nankai University, Tianjin, 300071, P. R. China

    Dong Ni Tan

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  1. Dong Ni Tan
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Correspondence to Dong Ni Tan.

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Supported by the Fundamental Research Funds for the Central Universities and National Natural Science Foundation of China (Grant No. 10871101)

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Tan, D.N. Extension of isometries on the unit sphere of L p spaces. Acta. Math. Sin.-English Ser. 28, 1197–1208 (2012). https://doi.org/10.1007/s10114-011-0302-6

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