JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2017, Vol. 52 ›› Issue (2): 44-48.doi: 10.6040/j.issn.1671-9352.0.2016.225

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Linear extension of isometries between the unit spheres of two-dimensional normed spaces

LIU Xiao-wei   

  1. Department of Mathematics, School of Science, Tianjin University, Tianjing 300350, China
  • Received:2016-05-19 Online:2017-02-20 Published:2017-01-18

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Abstract: We study the extension of isometries between the unit spheres S(E) and S(F), where E and F are both 2-dimensional real normed space. It is proved that surjective isometry V0 defined from S(E) to S(F) can be extended to be a linearly isometry V defined on the whole space S(E) under the some conditions.

Key words: Tingleys problem, isometry, linear extension

CLC Number: 

  • O177.2
[1] MAZUR S, ULAM S. Sur les transformationes isométriques despaces vectoriels normés[J]. C R Acad Sci Paris, 1932, 194: 946-948.
[2] MANKIEWICZ P. On extension of isometries in normed linear spaces[J]. Bull Acad Polon Sci Sér Sci Math Astronom Phys, 1972, 20(5):367-371.
[3] TINGLEY D. Isometries of the unit sphere[J]. Geom Dedicata, 1987, 22(3):371-378.
[4] DING Guanggui. The isometric extension problem in the unit spheres of lp(Γ)(p>1) type spaces[J]. Sci China Ser A, 2003, 46(3):333-338.
[5] DING Guanggui. The representation theorem of onto isometric mappings between two unit spheres of l∞-type spaces and the application on isometric extension problem[J]. Sci China Ser A, 2004, 47(5):722-729.
[6] DING Guanggui. The representation theorem of onto isometric mappings between two unit spheres of l1(Γ)type spaces and the application to the isometric extension problem[J]. Acta Math Sin Engl Series, 2004, 20(6):1089-1094.
[7] 方习年, 王建华. 单位球面间等距映射的线性延拓[J]. 数学学报, 2005, 48(6):1109-1112. FANG Xinian, WANG Jianhua. On linear extension of isometries between the unit sphere[J]. Acta Math Sin China Ser, 2005, 48(6):1109-1112.
[8] FANG Xinian, WANG Jianhua. On extension of isometries between the unit spheres of normed space E and C(Ω)[J]. Acta Math Sin Engl Ser, 2006, 22(6):1819-1824.
[9] TAN Dongni. On extension of isometries between unit spheres of Lp(μ)[J]. Acta Math Sin Engl Ser, 2012, 28(6):1197-1208.
[10] LIU Rui, ZHANG Lun. On extension of isometries and approximate isometries between unit spheres[J]. J Math Anal Appl, 2009, 352(2):749-761.
[11] DING Guanggui. The isometric extension of into mappings on unit spheres of AL-spaces[J]. Sci China Ser A, 2008, 51(10):1904-1918.
[12] CHENG Lixin, DONG Yunbai. On a generalized Mazur-Ulam question: extension of isometries between unit spheres of Banach spaces[J]. J Math Anal Appl, 2011, 377(2):464-470.
[13] KADETS V, MARTIN M. Extension of isometries between unit spheres of finite-dimensional polyhedral Banach spaces[J]. J Math Anal Appl, 2012, 396(2):441-447.
[14] 王瑞东. 二维严格凸赋范空间单位球面间等距映射的线性延拓[J]. 数学学报, 2008, 51(5):847-852. WANG Ruidong. Linear extension of isometries between the unit spheres of two-dimensional strictly convex normed spaces[J]. Acta Math Sin China Ser, 2008, 51(5):847-852.
[15] TANAKA R. Tingleys problem on symmetric absolute normalized norms on R2[J]. Acta Math Sin Engl Ser Aug, 2014, 30(8):1324-1340.
[16] ALONSO J, MARTÍN P. Moving triangles over a sphere[J]. Math Nachr, 2006, 279(16):1735-1738.
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