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NONEXPANSIVE MAPPINGS ON THE UNIT SPHERES OF SOME BANACH SPACES

Published online by Cambridge University Press:  19 June 2009

DONG-NI TAN*
Affiliation:
School of Mathematical Sciences, Nankai University, Tianjin 300071, PR China (email: 0110127@mail.nankai.edu.cn)
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Abstract

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We characterize surjective nonexpansive mappings between unit spheres of ℒ(Γ)-type spaces. We show that such mappings turn out to be isometries and can be extended to linear isometries on the whole space ℒ(Γ).

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2009

Footnotes

This work was partially supported by Research Foundation for Doctor Programme (20070055010) and National Natural Science Foundation of China (10571090).

References

[1] An, G. M., ‘Isometries on unit sphere of ( β n)’, J. Math. Anal. Appl. 301 (2005), 249254.Google Scholar
[2] Benyamini, Y. and Lindenstrauss, J., Geometric Nonlinear Functional Analysis, Vol. I, Colloquium Publications, Vol. 48 (American Mathematical Society, Providence, RI, 2000).Google Scholar
[3] Brown, T. A. and Comfort, W. W., ‘New method for expansion and contraction maps in uniform spaces’, Proc. Amer. Math. Soc. 11 (1960), 483486.Google Scholar
[4] Ding, G. G., ‘The 1-Lipschitz mapping between the unit spheres of two Hilbert spaces can be extended to a real linear linear isometry of the whole space’, Sci. China Ser. A 45(4) (2002), 479483.CrossRefGoogle Scholar
[5] Ding, G. G., ‘The isometric extension problem in the unit spheres of p(Γ)(p>1) type spaces’, Sci. China Ser. A 46(3) (2003), 333338.CrossRefGoogle Scholar
[6] Ding, G. G., ‘The representation theorem of onto isometric mapping between two unit spheres of 1(Γ) type spaces and the application on isometric extension problem’, Acta Math. Sin. (Engl. Ser.) 20(6) (2004), 10891094.Google Scholar
[7] Ding, G. G., ‘The representation of onto isometric mappings between two spheres of -type spaces and the application on isometric extension problem’, Sci. China Ser. A 34(2) (2004), 157164 (in Chinese); 47 (5) (2004), 722–729 (in English).Google Scholar
[8] Ding, G. G., ‘On the extension of isometries between unit spheres of E and C(Ω)’, Acta Math. Sin. (Engl. Ser.) 19(4) (2003), 793800.Google Scholar
[9] Ding, G. G., ‘The isometric extension of the into mapping from the ℒ(Γ)-type space to some normed space E’, Illinois J. Math. 52(2) (2007), 445453.Google Scholar
[10] Freudenthal, H. and Hurewicz, W., ‘Dehnungen, Verkurzungen, Isometrien’, Fund. Math. 26 (1936), 120122.Google Scholar
[11] Liu, R., ‘On extension of isometries between the unit spheres of ℒ-type space and some Banach space E’, J. Math. Anal. Appl. 333 (2007), 959970.Google Scholar
[12] Mankiewicz, P., ‘On extension of isometries in normed linear spaces’, Bull. Acad. Polon. Sci. 20 (1972), 367371.Google Scholar
[13] Rhodes, F., ‘A generalization of isometries to uniform spaces’, Proc. Cambridge Philos. Soc. 52 (1956), 399405.Google Scholar
[14] Tingley, D., ‘Isometries of the unit sphere’, Geom. Dedicata 22 (1987), 371378.Google Scholar
[15] Wang, J., ‘On extension of isometries between unit spheres of AL p-spaces (1<p<)’, Proc. Amer. Math. Soc. 132(10) (2004), 28992909.Google Scholar
[16] Yang, X. Z., ‘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 (2006), 985992.Google Scholar