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.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Al-Halees, H., Fleming, R.: On 2-local isometries on continuous vector valued function spaces. J. Math. Anal. Appl. 354, 70–77 (2009)
Apostol, T.M.: Mathematical Analysis, 2nd edn. Addison-Wesley, Reading (1974)
Araujo, J.: Linear isometries between spaces of functions of bounded variation. Bull. Aust. Math. Soc. 59, 335–341 (1999)
Győry, M.: 2-local isometries of \(C_0(X)\). Acta Sci. Math. (Szeged) 67, 735–746 (2001)
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)
Hatori, O., Oi, S.: 2-local isometries on function spaces. Contemp. Math. 737, 89–106 (2019)
Hosseini, M.: Algebraic reflexivity of sets of bounded linear operators on absolutely continuous function spaces. Oper. Matrices 13(3), 887–905 (2019)
Hosseini, M.: Generalized 2-local isometries of spaces of continuously differentiable functions. Quaest. Math. 40(8), 1003–1014 (2017)
Hosseini, M.: Isometries on spaces of absolutely continuous vector-valued functions. J. Math. Anal. Appl. 463, 386–397 (2018)
Hosseini, M.: Real-linear isometries on spaces of functions of bounded variation. Results Math. 70, 299–311 (2016)
Jarosz, K.: Isometries in semisimple, commutative Banach algebras. Proc. Am. Math. Soc. 94(1), 65–71 (1985)
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)
Jiménez-Vargas, A., Villegas-Vallecillos, M.: 2-local isometries on spaces of Lipschitz functions. Can. Math. Bull. 54, 680–692 (2011)
Kowalski, S., Słodkowski, Z.: A characterization of multiplicative linear functionals in Banach algebras. Stud. Math. 67, 215–223 (1980)
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)
Molnár, L.: 2-local isometries of some operator algebras. Proc. Edinb. Math. 45, 349–352 (2002)
Pathak, V.D.: Linear isometries of spaces of absolutely continuous functions. Can. J. Math. 34(2), 298–306 (1982)
Šemrl, P.: Local automorphisms and derivations on B(H). Proc. Am. Math. Soc. 125, 2677–2680 (1997)
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
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
- Linear isometry
- 2-Local isometry
- Functions of bounded variation
- Absolutely continuous functions
Mathematics Subject Classification
- Primary 47B38
- Secondary 46J10