Extension of Isometries on the Unit Sphere of L^p Spaces  

作　　者：Dong Ni TAN 

机构地区：[1]Department of Mathematics,Tianjin University of Technology [2]School of Mathematical Science,Nankai University

出　　处：《Acta Mathematica Sinica,English Series》2012年第6期1197-1208,共12页数学学报（英文版）

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

摘　　要：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-l-Lipschitz map between the unit spheres of L^p（μ1, H1） and L^p（μ2, H2） must be an isometry and can be extended to a linear isometry from L^p（μ2, H2） to L^p（μ2, H2）, where H1 and H2 are Hilbert spaces.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-l-Lipschitz map between the unit spheres of L^p（μ1, H1） and L^p（μ2, H2） must be an isometry and can be extended to a linear isometry from L^p（μ2, H2） to L^p（μ2, H2）, where H1 and H2 are Hilbert spaces.

关 键 词：Tingley＇s problem 1-Lipschitz anti-l-Lipschitz ISOMETRY isometric extension 

分 类 号：O177[理学—数学]