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Characterizations of Norm–Parallelism in Spaces of Continuous Functions

  • Ali Zamani
Original Paper

Abstract

In this paper, we consider the characterization of norm–parallelism problem in some classical Banach spaces. In particular, for two continuous functions fg on a compact Hausdorff space K, we show that f is norm–parallel to g if and only if there exists a probability measure (i.e., positive and of full measure equal to 1) \(\mu \) with its support contained in the norm-attaining set \(\{x\in K: \, |f(x)| = \Vert f\Vert \}\) such that \(\big |\int _K \overline{f(x)}g(x){\text {d}}\mu (x)\big | = \Vert f\Vert \,\Vert g\Vert \).

Keywords

Norm–parallelism Banach space of continuous functions Probability measure 

Mathematics Subject Classification

47A30 46B20 46E15 

Notes

Acknowledgements

The author would like to thank the referees for their careful reading of the manuscript and useful comments.

References

  1. 1.
    Arambašić, Lj, Rajić, R.: A strong version of the Birkhoff–James orthogonality in Hilbert \(C^*\)-modules. Ann. Funct. Anal. 5(1), 109–120 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Barraa, M., Boumazgour, M.: Inner derivations and norm equality. Proc. Am. Math. Soc. 130(2), 471–476 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bottazzi, T., Conde, C., Moslehian, M.S., Wójcik, P., Zamani, A.: Orthogonality and parallelism of operators on various Banach spaces. J. Aust. Math. Soc. (to appear) Google Scholar
  4. 4.
    Chica, M., Kadets, V., Martín, M., Moreno-Pulido, S., Rambla-Barreno, F.: Bishop–Phelps–Bollobás moduli of a Banach space. J. Math. Anal. Appl. 412(2), 697–719 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chmieliński, J., Łukasik, R., Wójcik, P.: On the stability of the orthogonality equation and the orthogonality-preserving property with two unknown functions. Banach J. Math. Anal. 10(4), 828–847 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ghosh, P., Sain, D., Paul, K.: On symmetry of Birkhoff-James orthogonality of linear operators. Adv. Oper. Theory 2(4), 428–434 (2017)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Grover, P.: Orthogonality of matrices in the Ky Fan \(k\)-norms. Linear Multilinear Algebra 65(3), 496–509 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Keckić, D.: Orthogonality and smooth points in \(C(K)\) and \(C_b(\Omega )\). Eurasian Math. J. 3(4), 44–52 (2012)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Moslehian, M.S., Zamani, A.: Mappings preserving approximate orthogonality in Hilbert \(C^*\)-modules. Math Scand. 122, 257–276 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Rudin, W.: Functional Analysis. McGraw-Hill Inc, New York (1973)zbMATHGoogle Scholar
  11. 11.
    Seddik, A.: Rank one operators and norm of elementary operators. Linear Algebra Appl. 424, 177–183 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Seddik, A.: On the injective norm of \(\Sigma_{i=1}^n A_i\otimes B_i\) and characterization of normaloid operators. Oper. Matrices 2(1), 67–77 (2008)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Werner, D.: The Daugavet equation for operators on function spaces. J. Funct. Anal. 143, 117–128 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Wójcik, P.: Norm-parallelism in classical \(M\)-ideals. Indag. Math. (N.S.) 28(2), 287–293 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Zamani, A.: The operator-valued parallelism. Linear Algebra Appl. 505, 282–295 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Zamani, A., Moslehian, M.S.: Exact and approximate operator parallelism. Can. Math. Bull. 58(1), 207–224 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Zamani, A., Moslehian, M.S.: Norm-parallelism in the geometry of Hilbert \(C^*\)-modules. Indag. Math. (N.S.) 27(1), 266–281 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Iranian Mathematical Society 2018

Authors and Affiliations

  1. 1.Department of MathematicsFarhangian UniversityTehranIran

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