Extension of isometries on the unit sphere of \(L^p\) spaces. (English) Zbl 1271.46011
The author gives a positive partial answer to Tingley’s isometric extension problem (i.e.,whether every onto isometry between unit spheres of Banach spaces can be extended to a (linear) isometry between the whole spaces). It is proved that every surjective isometry between the unit spheres of \(L_p(\mu)\) (\(1<p<\infty\), \(p\neq 2\)) and a Banach space \(E\) can be extended to a linear isometry from \(L_p(\mu)\) onto \(E\). Some results on extensions of \(1\)-Lipschitz and anti-\(1\)-Lipschitz maps between unit spheres are also presented.
More information on Tingley’s problem can be found in [D. Tingley, Geom. Dedicata 22, 371–378 (1987; Zbl 0615.51005)] and the recent survey G. G. Ding [Sci. China, Ser. A 52, No. 10, 2069–2083 (2009; Zbl 1190.46013)].
More information on Tingley’s problem can be found in [D. Tingley, Geom. Dedicata 22, 371–378 (1987; Zbl 0615.51005)] and the recent survey G. G. Ding [Sci. China, Ser. A 52, No. 10, 2069–2083 (2009; Zbl 1190.46013)].
Reviewer: Miguel Martín (Granada)
MSC:
| 46B04 | Isometric theory of Banach spaces |
| 46B20 | Geometry and structure of normed linear spaces |
| 46B25 | Classical Banach spaces in the general theory |
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