Abstract
In 2000 Victor Lomonosov suggested a counterexample to the complex version of the Bishop–Phelps theorem on modulus support functionals. We discuss the \(c_0\)-analog of that example and demonstrate that the set of sup-attaining functionals is non-trivial, thus answering an open question, asked in Kadets et al. (The mathematical legacy of Victor Lomonosov. Operator theory. Advances in analysis and geometry 2. De Gruyter, Berlin, 157–187, 2020).
Similar content being viewed by others
References
Albiac, F., Kalton, N.: Topics in Banach space theory. Graduate Texts in Mathematics 233. Berlin: Springer. xi, 373 p. (2006)
Bishop, E., Phelps, R.R.: A proof that every Banach space is subreflexive. Bull. Am. Math. Soc. 67, 97–98 (1961)
Bishop, E., Phelps, R.R.: The support functionals of a convex set. Proc. Symp. Pure Math. 7, 27–35 (1963)
Havin, V.P., Jöricke, B.: The uncertainty principle in harmonic analysis. Springer-Verlag, Berlin (1994)
Helson, H.: Harmonic analysis. Hindustan Book Agency, New Delhi (2010)
Kadets, V.: A course in functional analysis and measure theory. Translated from the Russian by Andrei Iacob. Universitext. Cham: Springer. xxii, 539 p. (2018)
Kadets, V., Lopez, G., Martín, M., Werner, D.: Norm attaining operators of finite rank. In: Aron, Richard M.; Gallardo Gutiérrez, Eva A.; Martin, Miguel; Ryabogin, Dmitry; Spitkovsky, Ilya M.; Zvavitch, Artem (editors). The mathematical legacy of Victor Lomonosov. Operator theory. Advances in Analysis and Geometry 2. Berlin: De Gruyter, 300 p. (2020), 157–187
Koosis, P.: Introduction to \(H_p\) spaces. CUP, Cambridge (1980)
Lindenstrauss, J., Tzafriri, L.: Classical banach spaces I: sequence spaces. Springer, Berlin (1977)
Lomonosov, V.: A counterexample to the Bishop–Phelps theorem in complex spaces. Israel J. Math. 115, 25–28 (2000)
Lomonosov, V.: On the Bishop–Phelps theorem in complex spaces. Quaest. Math. 23, 187–191 (2000)
Lomonosov, V.: The Bishop–Phelps theorem fails for uniform non-selfadjoint dual operator algebras. J. Funct. Anal. 185, 214–219 (2001)
Noell, A.: Peak sets and boundary interpolation sets for the unit disc: a survey, Bull. London Math. Soc. (2020) https://doi.org/10.1112/blms.12414
Noell, A., Wolff, T.: On peak sets for Lip \(\alpha \) classes. J. Funct. Anal. 86, 136–179 (1989)
Phelps, R.R.: The Bishop–Phelps theorem in complex spaces: an open problem. Lect. Notes Pure Appl. Math. 136, 337–340 (1991)
Simon, B.: A comprehensive course in analysis. Part 4:Operator Theory, AMS, Providence, RI, 2015
Zygmund, A.: Trigonometric series, 3d ed., Cambridge Math. Library, CUP, Cambridge, (2002)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The research of the second author was done in the framework of the Ukrainian Ministry of Education and Science Research Program 0118U002036, was partially supported by the project PGC2018-093794-B-I00 (MCIU/AEI/FEDER, UE) and on the final stage by the National Research Foundation of Ukraine funded by Ukrainian State budget in frames of Project 2020.02/0096 “Operators in infinite-dimensional spaces: the interplay between geometry, algebra and topology”
Rights and permissions
About this article
Cite this article
Golinskii, L., Kadets, V. Modulus support functionals, Rajchman measures and peak functions. RACSAM 115, 52 (2021). https://doi.org/10.1007/s13398-020-00974-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13398-020-00974-5