单位球面上的Figiel型问题

The Figiel type problem on unit spheres

本文研究Banach空间单位球面间等距嵌入的Figiel型问题.首先通过一个反例表明经典的Figiel定理不能平凡地推广到球面间等距嵌入的情形.然后,找出一个自然的必要条件,并进一步证明:在这一条件下,从几类具体空间(包括C(Ω)、L^(1)(μ)和L^(∞)(Γ)-型空间)的单位球面映到另一个Banach空间单位球面的等距嵌入都有相应的Figiel型定理.最后,得到本文所研究的单位球面上的Figiel型问题与Tingley问题的紧密联系.

In this paper,we study the Figiel type problem of isometries between the unit spheres of Banach spaces.First,by a counterexample,we show that the classical Figiel’s theorem cannot be trivially generalized to the case of isometric embeddings on unit spheres.Then we find a natural necessary condition to ensure the corresponding Figiel type theorem.Under this natural condition,we prove the Figiel type theorems of isometries from the unit sphere of one specific space(C(Ω),L^(1)(μ)or L^(∞)(Γ)-type spaces)into the unit sphere of another Banach space.Namely,the isometry owns a linear norm-1 operator as a left-inverse.Finally,we obtain the relationship between the Tingley problem and the Figiel type problem on unit spheres.