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 Figie...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].展开更多
A pair of Banach spaces (X, Y) is said to be stable if for every ￡-isometry f : X→Y,there exist γ> 0 and a bounded linear operator T : L(f)→X with ||T||≤α such that ||Tf(x)- x||≤γε for all x ∈ X, where L(...A pair of Banach spaces (X, Y) is said to be stable if for every ￡-isometry f : X→Y,there exist γ> 0 and a bounded linear operator T : L(f)→X with ||T||≤α such that ||Tf(x)- x||≤γε for all x ∈ X, where L(f) is the closed linear span of f(X). In this article, we study the stability of a pair of Banach spaces (X, Y) when X is a C(K) space. This gives a new positive answer to Qian's problem. Finally, we also obtain a nonlinear version for Qian's problem.展开更多
基金the National Nature Science Foundation of China(11671214,11971348,12071230)the Hundred Young Academia Leaders Program of Nankai University(63223027,ZB22000105)+1 种基金the Undergraduate Education and Teaching Project of Nankai University(NKJG2022053)the National College Students’Innovation and Entrepreneurship Training Program of Nankai University(202210055048)。
文摘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].
基金supported in part by NSFC(11601264,11471270 and 11471271)the Fundamental Research Funds for the Central Universities(20720160037)+4 种基金the Outstanding Youth Scientific Research Personnel Training Program of Fujian Provincethe High level Talents Innovation and Entrepreneurship Project of Quanzhou City(2017Z032)the Research Foundation of Quanzhou Normal University(2016YYKJ12)the Natural Science Foundation of Fujian Province of China(2019J05103)supported in part by NSFC(11628102)
文摘A pair of Banach spaces (X, Y) is said to be stable if for every ￡-isometry f : X→Y,there exist γ> 0 and a bounded linear operator T : L(f)→X with ||T||≤α such that ||Tf(x)- x||≤γε for all x ∈ X, where L(f) is the closed linear span of f(X). In this article, we study the stability of a pair of Banach spaces (X, Y) when X is a C(K) space. This gives a new positive answer to Qian's problem. Finally, we also obtain a nonlinear version for Qian's problem.
