Skip to main content
SpringerLink
Log in

Maps Preserving the Norm of the Positive Sum in Lp Spaces

  • Published:
Acta Mathematica Scientia Aims and scope Submit manuscript

Abstract

For 1 < p < ∞, let S(Lp)+ be the set of positive elements in Lp with norm one. Assume that V0: S(Lp(Ω1))+ → S(Lp (Ω2))+ is a surjective norm-additive map; that is,

$$\left\| {{V_0}(x) + {V_0}(y)} \right\| = \left\| {x + y} \right\|,\,\,\,\,\forall x,y \in S{({L_p}({\Omega _1}))_{ + \cdot }}$$

In this paper, we show that V0 can be extended to an isometry from Lp(Ω1) onto Lp (Ω2).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Fleming R J, Jamison J E. Isometries on Banach spaces: Function spaces (Vol 1). Boca Raton: CRC Press, 2003

    MATH  Google Scholar 

  2. Fleming R J, Jamison J E. Isometries on Banach spaces: Vector-valued Function spaces and Operator Spaces (Vol 2). Boca Raton: CRC Press, 2007

    Book  Google Scholar 

  3. Mazur S, Ulam S. Sur les transformations isometriques d’espaces vectoriels normes. C R Acad Sci Paris, 1932, 194: 946–948

    MATH  Google Scholar 

  4. Molnár L. Selected preserve problems on algebraic structures of linear operators and on function spaces. Berlin: Springer-Verlag, 2007

    MATH  Google Scholar 

  5. Molnár L, Szokol P. Transformations preserving norms of means of positive operators and nonnegative functions. Integr Equ Oper Theory, 2015, 83: 271–290

    Article  MathSciNet  Google Scholar 

  6. Tonev T, Yates R. Norm-linear and norm-additive operators between uniform algebras. J Math Anal Appl, 2009, 357: 45–53

    Article  MathSciNet  Google Scholar 

  7. Zhang J, Tsai M-C, Wong N-C. Norm of positive sum preservers of smooth Banach lattices and Lp(µ) spaces. J Nonlinear and Convex Anal, 2019, 20(12): 2613–2621

    MathSciNet  MATH  Google Scholar 

  8. Zhang J, Tsai M-C, Wong N-C. Norm of positive sum preservers of non-commutative Lp(\({\cal M}\)) spaces. J Nonlinear and Convex Anal, 2021, 22(2): 251–264

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Nankai University, Tianjin, 300071, China

    Jingjing Hao  (郝晶晶)

  2. Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan, 430062, China

    Yunbai Dong  (董云柏)

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

    Lei Li  (李蠢)

Authors
  1. Jingjing Hao  (郝晶晶)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yunbai Dong  (董云柏)
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lei Li  (李蠢)
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yunbai Dong  (董云柏).

Additional information

Dong is partially supported by the NSF of China (11671314). Li is partially supported by the NSF of China (12171251).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hao, J., Dong, Y. & Li, L. Maps Preserving the Norm of the Positive Sum in Lp Spaces. Acta Math Sci 42, 789–794 (2022). https://doi.org/10.1007/s10473-022-0223-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

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

Key words

2010 MR Subject Classification

Search

Navigation