ON THE FIGIEL TYPE PROBLEM AND EXTENSION OFε-ISOMETRY BETWEEN UNIT SPHERES  

作　　者：刘锐 尹际富 Rui LIU;Jifu YIN(School of Mathematical Sciences and LPMC,Nankai University,Tianjin 300071,China)

机构地区：[1]School of Mathematical Sciences and LPMC,Nankai University,Tianjin 300071,China

出　　处：《Acta Mathematica Scientia》2023年第4期1503-1517,共15页数学物理学报（B辑英文版）

基　　金：the National Nature Science Foundation of China(11671214,11971348,12071230);the Hundred Young Academia Leaders Program of Nankai University(63223027,ZB22000105);the Undergraduate Education and Teaching Project of Nankai University(NKJG2022053);the National College Students’Innovation and Entrepreneurship Training Program of Nankai University(202210055048)。

摘　　要：This paper studies two isometric problems between unit spheres of Banach spaces.In the first part,we introduce and study the Figiel type problem of isometric embeddings between unit spheres.However,the classical Figiel theorem on the whole space cannot be trivially generalized to this case,and this is pointed out by a counterexample.After establishing this,we find a natural necessary condition required by the existence of the Figiel operator.Furthermore,we prove that when X is a space with the T-property,this condition is also sufficient for an isometric embedding T:S_(X)→S_(Y) to admit the Figiel operator.This answers the Figiel type problem on unit spheres for a large class of spaces.In the second part,we consider the extension of bijectiveε-isometries between unit spheres of two Banach spaces.It is shown that every bijectiveε-isometry between unit spheres of a local GL-space and another Banach space can be extended to be a bijective 5ε-isometry between the corresponding unit balls.In particular,whenε=0,this recovers the MUP for local GL-spaces obtained in[40].

关 键 词：unit spheres isometric embedding Figiel type problem ε-isometry EXTENSION 

分 类 号：O177.2[理学—数学]