On isometric extension problem between two unit spheres

Science China Mathematics - Tập 52 - Trang 2069-2083 - 2009
GuangGui Ding1
1School of Mathematical Sciences and LPMC, Nankai University, Tianjin, China

Tóm tắt

In this paper we introduce the isometric extension problem of isometric mappings between two unit spheres. Some important results of the related problems are outlined and the recent progress is mentioned.

Tài liệu tham khảo

Tingley D. Isometries of the unit spheres. Geom Dedicata, 22: 371–378 (1987) Ding G. The 1-Lipschitz mapping between the unit spheres of two Hilbert spaces can be extended to a real linear isometry of the whole space. Sci China Ser A, 45(4): 479–483 (2002) Ding G. The representation theorem of onto isometric mappings between two unit spheres of l 1(Γ) type spaces and the application to the isometric extension problem. Acta Math Sin Engl Ser, 20(6): 1089–1094 (2004) Ding G. The isometric extension problem in the unit spheres of l p(Γ) (p > 1) type spaces. Sci China Ser A, 32(11): 991–995 (2002) Ding G. The representation of onto isometric mappings between two spheres of l ∞ type spaces and the application on isometric extension problem. Sci China Ser A, 34(2): 157–164 (2004) Banach S. Theorié Des Opérations Linéaires. Warszawa Monografie Matematyczne, 1932 An G. Isometries on unit spheres of (l β n). J Math Anal Appl, 301: 249–254 (2005) Wang R, Orihara A. Isometries between the unit spheres of l 1-sum of strictly convex normed spaces. Acta Sci Nat Univ Nankai, 35(1): 38–42 (2002) Liu R. Isometries between the unit spheres of l 1-sum of strictly convex normed spaces. Acta Math Sinica Chin Ser, 50(1): 228–232 (2007) Zhang L. On the isometric extension problem from the unit sphere S(l ∞(2) ) into S(l ∞(3) ). Acta Sci Nat Univ Nankai, 39(2): 110–112 (2006) Wang J. On extension of isometries between unit spheres of AL p-spaces (1 < p < ∞). Proc Amer Math Soc, 132(10): 2899–2909 (2004) Hou Z. The isometric extension of the into mapping between the unit spheres of AL p-spaces (1 < p < ∞). Acta Math Sinica Chin Ser, 50(6): 1435–1440 (2007) Lindenstrauss J, Tzafriri L. Classical Banach Spaces II: Function Spaces. Ergebnisse 92. Berlin-Heidelberg-New York: Spring-Verlag, 1979 Yang X. On extension of isometries between unit spheres of L p(μ) and L p(ν,H) (1 < p ≠ 2, H is a Hilbert space). J Math Anal Appl, 323: 985–992 (2006) Ding G. On the extension of isometries between unit spheres of E and C(Ω). Acta Math Sin Engl Ser, 19(4): 793–800 (2003) Fang X, Wang J. On linear extension of isometries between the unit spheres. Acta Math Sinica Chin Ser, 48(6): 1109–1112 (2005) Fang X, Wang J. Extension of isometries between the unit spheres of normed space E and C(Ω). Acta Math Sin Engl Ser, 22(6): 1819–1824 (2006) Fu X. The isometric extension of the into mapping from the unit sphere S(E) to S(l ∞(Γ)). Acta Math Sin Engl Ser, 24(9): 1475–1482 (2008) Fang X, Wang J. Extension of isometries between unit spheres of normed space E and l 1(Γ). Acta Math Sinica Chin Ser, 51(1): 24–28 (2008) Ding G. Extension of isometries on the unit sphere of AL-space. Sci China Ser A, 38(5): 541–555 (2008) Ding G. The isometric extension of the into mapping from a L ∞(Γ)-type space to some Banach space. Illinois J Math, 51(2): 445–453 (2007) Liu R. On extension of isometries between unit spheres of L ∞(Γ)-type space and a Banach space E. J Math Anal Appl, 333: 959–970 (2007) Ding G. The isometrically linear extensions of into-mappings between two unit spheres of the Asplund generated spaces. Submitted Fang X. On extension of 1-Lipschitz mappings between two unit spheres of l p(Γ) type spaces (1 < p < ∞). J Math Research and Exposition, to appear, 2009 Liu R. 1-Lipschitz mappings between unit spheres of Banach spaces. Acta Math Sinica Chin Ser, 50(5): 1064–1070 (2007) Ding G. On linearly isometric extensions for 1-Lipschitz mappings between unit spheres of AL p-spaces (p > 2). Acta Math Sin Engl Ser, to appear, 2009 Wang R. On Extension of 1-Lipschitz Mappings between l 1(Γ) type spaces. Acta Sci Nat Univ Nankai, to appear, 2009 Tan D. Nonexpansive mappings on the unit spheres of some Banach spaces. Bull Aust Math Soc, 80: 139–146 (2009) Mazur S, Ulam S. Sur less transformations isometriques d’espaces vectoriels normés. C R Math Acad Sci Paris, 194: 946–948 (1932) Day M M. Normed Linear Spaces. Berlin-Heidelberg-New York: Springer-Verlag, 1973 Li L, Ren W Y. On extension of isometries between unit spheres of L ∞ and E. Quaest Math, 31: 209–218 (2008)