Extension of isometries on the unit sphere of L p spaces

Springer Science and Business Media LLC - Tập 28 - Trang 1197-1208 - 2011
Dong Ni Tan1,2
1Department of Mathematics, Tianjin University of Technology, Tianjin, P.R. China
2School of Mathematical Science, Nankai University, Tianjin, P. R. China

Tóm tắt

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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