2-Local isometries between spaces of functions of bounded variation

Abstract

Given two subsets X and Y of the real line with at least two points, we apply results on surjective linear isometries between Banach spaces of all functions of bounded variation BV(X) and BV(Y) to show that every 2-local isometry \(T:BV(X)\longrightarrow BV(Y)\) is a constant multiple of an isometric linear algebra isomorphism. Moreover, similar results are given for the closed subspaces of BV(X) and BV(Y) consisting of all continuous (resp. absolutely continuous) functions when X and Y are compact.

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References

  1. 1.

    Al-Halees, H., Fleming, R.: On 2-local isometries on continuous vector valued function spaces. J. Math. Anal. Appl. 354, 70–77 (2009)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Apostol, T.M.: Mathematical Analysis, 2nd edn. Addison-Wesley, Reading (1974)

    MATH  Google Scholar 

  3. 3.

    Araujo, J.: Linear isometries between spaces of functions of bounded variation. Bull. Aust. Math. Soc. 59, 335–341 (1999)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Győry, M.: 2-local isometries of \(C_0(X)\). Acta Sci. Math. (Szeged) 67, 735–746 (2001)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Hatori, O., Miura, T., Oka, H., Takagi, H.: 2-local isometries and 2-local automorphisms on uniform algebras. Int. Math. Forum 2(50), 2491–2502 (2007)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Hatori, O., Oi, S.: 2-local isometries on function spaces. Contemp. Math. 737, 89–106 (2019)

  7. 7.

    Hosseini, M.: Algebraic reflexivity of sets of bounded linear operators on absolutely continuous function spaces. Oper. Matrices 13(3), 887–905 (2019)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Hosseini, M.: Generalized 2-local isometries of spaces of continuously differentiable functions. Quaest. Math. 40(8), 1003–1014 (2017)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Hosseini, M.: Isometries on spaces of absolutely continuous vector-valued functions. J. Math. Anal. Appl. 463, 386–397 (2018)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Hosseini, M.: Real-linear isometries on spaces of functions of bounded variation. Results Math. 70, 299–311 (2016)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Jarosz, K.: Isometries in semisimple, commutative Banach algebras. Proc. Am. Math. Soc. 94(1), 65–71 (1985)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Jiménez-Vargas, A., Li, L., Peralta, A.M., Wang, L., Wang, Y.-S.: 2-local standard isometries on vector-valued Lipschitz function spaces. J. Math. Anal. Appl. 461, 1287–1298 (2018)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Jiménez-Vargas, A., Villegas-Vallecillos, M.: 2-local isometries on spaces of Lipschitz functions. Can. Math. Bull. 54, 680–692 (2011)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Kowalski, S., Słodkowski, Z.: A characterization of multiplicative linear functionals in Banach algebras. Stud. Math. 67, 215–223 (1980)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Li, L., Peralta, A.M., Wang, L., Wang, Y.-S.: Weak-2-local isometries on uniform algebras and Lipschitz algebras. Publ. Mat. 63, 241–264 (2019)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Molnár, L.: 2-local isometries of some operator algebras. Proc. Edinb. Math. 45, 349–352 (2002)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Pathak, V.D.: Linear isometries of spaces of absolutely continuous functions. Can. J. Math. 34(2), 298–306 (1982)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Šemrl, P.: Local automorphisms and derivations on B(H). Proc. Am. Math. Soc. 125, 2677–2680 (1997)

    MathSciNet  Article  Google Scholar 

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Correspondence to Maliheh Hosseini.

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Hosseini, M. 2-Local isometries between spaces of functions of bounded variation. Positivity 24, 1101–1109 (2020). https://doi.org/10.1007/s11117-019-00721-0

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Keywords

  • Linear isometry
  • 2-Local isometry
  • Functions of bounded variation
  • Absolutely continuous functions

Mathematics Subject Classification

  • Primary 47B38
  • Secondary 46J10
  • 47B33