MAPS PRESERVING THE NORM OF THE POSITIVE SUM IN Lp SPACES

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MAPS PRESERVING THE NORM OF THE POSITIVE SUM IN L_p SPACES

MAPS PRESERVING THE NORM OF THE POSITIVE SUM IN L_p SPACES

MAPS PRESERVING THE NORM OF THE POSITIVE SUM IN L_p SPACES

MAPS PRESERVING THE NORM OF THE POSITIVE SUM IN L_p SPACES

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doi:10.1007/s10473-022-0223-8

数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (2): 789-794.doi: 10.1007/s10473-022-0223-8

郝晶晶¹, 董云柏², 李磊³

1. School of Mathematical Sciences, Nankai University, Tianjin 300071, China;
2. Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan 430062, China;
3. School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China

收稿日期:2020-12-24 修回日期:2021-03-11 出版日期:2022-04-25 发布日期:2022-04-22
通讯作者: Yunbai DONG,E-mail:baiyunmu301@126.com E-mail:baiyunmu301@126.com
作者简介:Jingjing HAO,E-mail:271869637@qq.com;Lei LI,E-mail:leilee@nankai.edu.cn
基金资助:
Dong is partially supported by the NSF of China (11671314). Li is partially supported by the NSF of China (12171251).

Jingjing HAO¹, Yunbai DONG², Lei LI³

1. School of Mathematical Sciences, Nankai University, Tianjin 300071, China;
2. Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan 430062, China;
3. School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China

Received:2020-12-24 Revised:2021-03-11 Online:2022-04-25 Published:2022-04-22
Supported by:
Dong is partially supported by the NSF of China (11671314). Li is partially supported by the NSF of China (12171251).

摘要： {For $1 < p < \infty$ , let $S(L_p)_+$ be the set of positive elements in $L_p$ with norm one. Assume that $V_0: S(L_p(\Omega_1))_{+}\to S(L_p(\Omega_2))_{+}$ is a surjective norm-additive map; that is,

‖ V_{0} (x) + V_{0} (y) ‖ = ‖ x + y ‖, \forall x, y \in S (L_{p} (Ω_{1}))_{+} .

$\|V_0(x)+V_0(y)\|=\|x+y\|,\quad\forall\,x, y\in S(L_p(\Omega_1 ))_{+}.$ In this paper, we show that

V_{0}

$V_0$ can be extended to an isometry from

L_{p} (Ω_{1})

$L_p(\Omega_1)$ onto

L_{p} (Ω_{2})

$L_p(\Omega_2)$ .

关键词: Norm-additive mappings, positive cones, L_p spaces

Abstract: {For $1 < p < \infty$ , let $S(L_p)_+$ be the set of positive elements in $L_p$ with norm one. Assume that $V_0: S(L_p(\Omega_1))_{+}\to S(L_p(\Omega_2))_{+}$ is a surjective norm-additive map; that is,

‖ V_{0} (x) + V_{0} (y) ‖ = ‖ x + y ‖, \forall x, y \in S (L_{p} (Ω_{1}))_{+} .

$\|V_0(x)+V_0(y)\|=\|x+y\|,\quad\forall\,x, y\in S(L_p(\Omega_1 ))_{+}.$ In this paper, we show that

V_{0}

$V_0$ can be extended to an isometry from

L_{p} (Ω_{1})

$L_p(\Omega_1)$ onto

L_{p} (Ω_{2})

$L_p(\Omega_2)$ .

Key words: Norm-additive mappings, positive cones, L_p spaces

中图分类号:

46E30

郝晶晶, 董云柏, 李磊. MAPS PRESERVING THE NORM OF THE POSITIVE SUM IN L_p SPACES[J]. 数学物理学报(英文版), 2022, 42(2): 789-794.

Jingjing HAO, Yunbai DONG, Lei LI. MAPS PRESERVING THE NORM OF THE POSITIVE SUM IN L_p SPACES[J]. Acta mathematica scientia,Series B, 2022, 42(2): 789-794.

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