空间Lp(Γ,∑,μ)(1＜p＜∞)和Banach空间E的单位球面之间等距算子的延拓

摘要

In this paper, we show the following result which generalizes some theorems (see [16] [18]): Let E be a normed space, V° :S(Lp((Γ,Σ,μ))→S(E)be a isometric mapping. If V° satisfies the following two hypotheses:( ⅰ )For all n∈[-1, 1]and Xaκ ∈ X(Γ), 1≤κ≤n, such that‖∑k=1,nξκμ(Ai)1/pV0[Xai/μ(Ai)1/p]‖p=∑k=1,n∣ξκ∣pμ(Ai),(ⅱ)For every f1, f2 ∈ S(Lp(Γ, ∑, μ)) and ξ1, ξ2 ∈[-1,1],such that‖ξ1V0(f1) + ξ2V0(f2) ‖ =1 ?∣ ξ1V0(f1) + ξ2V0(f2) ∈ V0[S(Lp(Γ, ∑, μ)], then V0 can be extended to a linear isometry defined on the whole space LP(Γ, ∑, μ).%文章得到以下结果(它改进了文献[16][18]中的一些结果):设E是一个赋范空间,V0是单位球面S(Lp(Γ,∑,μ))到单位球面S(E)内的等距映射.如果V0满足下列两个条件:(i)对于任意的自然数n,实数εk∈[-1,1]及xAk∈x(Γ),1≤k≤n,有‖n∑k=1ξkμ(At)1/pV0[xAt/μ(Ai)1/p]‖p=n∑k=1│ξk│pμ(Ai),(ii)对于任意的f1,f2∈S(Lp(Γ,∑,μ))和实数ξ1,ξ2∈[-1,1],有‖ξ1 V0(f1)+ξ2V0(f2)‖=1(→)│ξ1V0(f1)+ξ2V0(f2)∈V0[S(Lp(Γ,∑,μ)],那么V0可延拓为全空间Lp(Γ,∑,μ)上的等距线性算子.