MAPS PRESERVING THE NORM OF THE POSITIVE SUM IN Lp SPACES

Jingjing HAO; Yunbai DONG; Lei LI

doi:10.1007/s10473-022-0223-8

Acta mathematica scientia, Series B >

2022 , Vol. 42 >Issue 2: 789 - 794

DOI: https://doi.org/10.1007/s10473-022-0223-8

Articles

MAPS PRESERVING THE NORM OF THE POSITIVE SUM IN L_p SPACES

Jingjing HAO ,
Yunbai DONG ,
Lei LI

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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 date: 2020-12-24

Revised date: 2021-03-11

Online 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).

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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, \[\|V_0(x)+V_0(y)\|=\|x+y\|,\quad\forall\,x, y\in S(L_p(\Omega_1 ))_{+}.\] In this paper, we show that $V_0$ can be extended to an isometry from $L_p(\Omega_1)$ onto $L_p(\Omega_2)$.

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

Cite this article

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 . DOI: 10.1007/s10473-022-0223-8

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