ON THE FIGIEL TYPE PROBLEM AND EXTENSION OF $\varepsilon$-ISOMETRY BETWEEN UNIT SPHERES∗

doi:10.1007/s10473-023-0403-1

数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (4): 1503-1517.doi: 10.1007/s10473-023-0403-1

ON THE FIGIEL TYPE PROBLEM AND EXTENSION OF $\varepsilon$-ISOMETRY BETWEEN UNIT SPHERES^∗

Rui LIU, Jifu YIN^†

School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China

收稿日期:2022-04-08 修回日期:2022-09-22 发布日期:2023-08-08
作者简介:Rui LIU, E-mails: ruiliu@nankai.edu.cn
基金资助:
* 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) and the National College Students' Innovation and Entrepreneurship Training Program of Nankai University (202210055048).

ON THE FIGIEL TYPE PROBLEM AND EXTENSION OF $\varepsilon$-ISOMETRY BETWEEN UNIT SPHERES^∗

Rui LIU, Jifu YIN^†

School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China

Received:2022-04-08 Revised:2022-09-22 Published:2023-08-08
About author:Rui LIU, E-mails: ruiliu@nankai.edu.cn
Supported by:
* 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) and the National College Students' Innovation and Entrepreneurship Training Program of Nankai University (202210055048).

摘要/Abstract

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 $T: S_X\rightarrow S_Y$ 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 bijective $\varepsilon$-isometries between unit spheres of two Banach spaces. It is shown that every bijective $\varepsilon$-isometry between unit spheres of a local GL-space and another Banach space can be extended to be a bijective $5\varepsilon$-isometry between the corresponding unit balls. In particular, when $\varepsilon=0$, this recovers the MUP for local GL-spaces obtained in [40].

关键词: unit spheres, isometric embedding, Figiel type problem, $\varepsilon$-isometry, extension

Abstract: 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 $T: S_X\rightarrow S_Y$ 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 bijective $\varepsilon$-isometries between unit spheres of two Banach spaces. It is shown that every bijective $\varepsilon$-isometry between unit spheres of a local GL-space and another Banach space can be extended to be a bijective $5\varepsilon$-isometry between the corresponding unit balls. In particular, when $\varepsilon=0$, this recovers the MUP for local GL-spaces obtained in [40].

Key words: unit spheres, isometric embedding, Figiel type problem, $\varepsilon$-isometry, extension

Rui LIU, Jifu YIN. ON THE FIGIEL TYPE PROBLEM AND EXTENSION OF $\varepsilon$-ISOMETRY BETWEEN UNIT SPHERES^∗[J]. 数学物理学报(英文版), 2023, 43(4): 1503-1517.

Rui LIU, Jifu YIN. ON THE FIGIEL TYPE PROBLEM AND EXTENSION OF $\varepsilon$-ISOMETRY BETWEEN UNIT SPHERES^∗[J]. Acta mathematica scientia,Series B, 2023, 43(4): 1503-1517.

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ON THE FIGIEL TYPE PROBLEM AND EXTENSION OF $\varepsilon$-ISOMETRY BETWEEN UNIT SPHERES^∗

ON THE FIGIEL TYPE PROBLEM AND EXTENSION OF $\varepsilon$-ISOMETRY BETWEEN UNIT SPHERES^∗

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