Acta mathematica scientia,Series B ›› 2022, Vol. 42 ›› Issue (2): 789-794.doi: 10.1007/s10473-022-0223-8

• Articles • Previous Articles     Next Articles

MAPS PRESERVING THE NORM OF THE POSITIVE SUM IN Lp SPACES

Jingjing HAO1, Yunbai DONG2, Lei LI3   

  1. 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).

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, Lp spaces

CLC Number: 

  • 46E30
Trendmd