Extension of Isometries Between the Unit Spheres of Lp((Γ,Σ,μ)(1<p<∞)and a Banach Space E
Huang Zhong-jie · Journal of Yunnan Normal University · 2012
In this paper,we show the following result which generalizes some theorems(see):Let E be a normed space,V0:S(Lp((Γ,,μ))→S(E)be a isometric mapping.If V0 satisfies the following two hypotheses:(ⅰ)For all n∈and χAk∈χ(Γ),1≤k≤n,such that ‖sum from k=1 to n ξkμ(Ai)1/pV0〔(χAi)/(μ(Ai)1/p)〕‖p=sum from k=1 to n|ξk|pμ(Ai),(ⅱ)For every f1,f2∈S(Lp(Γ,Σ,μ))and ξ1,ξ2∈,such that ‖ξ1V0(f1)+ξ2V0(f2)‖=1|ξ1V0(f1)+ξ2V0(f2)∈V0[S(Lp(Γ,Σ,μ)], then V0can be extended to a linear isometry defined on the whole space Lp(Γ,Σ,μ).