Abstract
In this paper, we consider the characterization of norm–parallelism problem in some classical Banach spaces. In particular, for two continuous functions f, g 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 \(\big |\int _K \overline{f(x)}g(x){\text {d}}\mu (x)\big | = \Vert f\Vert \,\Vert g\Vert \).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arambašić, Lj, Rajić, R.: A strong version of the Birkhoff–James orthogonality in Hilbert \(C^*\)-modules. Ann. Funct. Anal. 5(1), 109–120 (2014)
Barraa, M., Boumazgour, M.: Inner derivations and norm equality. Proc. Am. Math. Soc. 130(2), 471–476 (2002)
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)
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)
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)
Ghosh, P., Sain, D., Paul, K.: On symmetry of Birkhoff-James orthogonality of linear operators. Adv. Oper. Theory 2(4), 428–434 (2017)
Grover, P.: Orthogonality of matrices in the Ky Fan \(k\)-norms. Linear Multilinear Algebra 65(3), 496–509 (2017)
Keckić, D.: Orthogonality and smooth points in \(C(K)\) and \(C_b(\Omega )\). Eurasian Math. J. 3(4), 44–52 (2012)
Moslehian, M.S., Zamani, A.: Mappings preserving approximate orthogonality in Hilbert \(C^*\)-modules. Math Scand. 122, 257–276 (2018)
Rudin, W.: Functional Analysis. McGraw-Hill Inc, New York (1973)
Seddik, A.: Rank one operators and norm of elementary operators. Linear Algebra Appl. 424, 177–183 (2007)
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)
Werner, D.: The Daugavet equation for operators on function spaces. J. Funct. Anal. 143, 117–128 (1997)
Wójcik, P.: Norm-parallelism in classical \(M\)-ideals. Indag. Math. (N.S.) 28(2), 287–293 (2017)
Zamani, A.: The operator-valued parallelism. Linear Algebra Appl. 505, 282–295 (2016)
Zamani, A., Moslehian, M.S.: Exact and approximate operator parallelism. Can. Math. Bull. 58(1), 207–224 (2015)
Zamani, A., Moslehian, M.S.: Norm-parallelism in the geometry of Hilbert \(C^*\)-modules. Indag. Math. (N.S.) 27(1), 266–281 (2016)
Acknowledgements
The author would like to thank the referees for their careful reading of the manuscript and useful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Hamid Reza Ebrahimi Vishki.
Rights and permissions
About this article
Cite this article
Zamani, A. Characterizations of Norm–Parallelism in Spaces of Continuous Functions. Bull. Iran. Math. Soc. 45, 557–567 (2019). https://doi.org/10.1007/s41980-018-0148-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s41980-018-0148-0