On continuous linear functionals in some spaces of functions analytic in disk

E. G. Rodikova

Ufa mathematical journal, Tome 15 (2023) no. 2, pp. 65-73

Résumé

The issue on description of linear continuous functionals on the spaces of analytic functions has been studied since the middle of 20th century. Historically, the structure of linear continuous functionals on the Hardy spaces $H^p$ for $p\geq 1$ was first established by A. Taylor in 1951. In the spaces $H^p$, $0$, this problem was solved by P. Duren, B. Romberg, A. Shields in 1969. We note that an estimate for the coefficient multipliers in these spaces was employed in the proof. In the present paper, by developing the method proposed in the work by P. Duren et al, we describe linear continuous functionals on area Privalov classes and Nevanlinna-Dzhrbashjan type spaces. The considered classes generalize the area Nevanlinna classes well-known in scientific literature. The idea of the proof of the main result is as follows: the issue on finding the general form of a continuous linear functional is reduced to finding a form of an arbitrary coefficient multiplier acting from a studied space into the space of bounded analytic functions. The latter problem in a simplified form can be formulated as follows: by what factors we should multiply the Taylor coefficients of the functions in a studied class in order to make them Taylor coefficients of some bounded analytic function.

Détail

Keywords: area Privalov spaces, Nevanlinna-Dzhrbashjan type spaces, linear continuous functionals
Mots-clés : coefficient multipliers.

Bibliographie

[1] V.I. Gavrilov, A.V. Subbotin, D.A. Efimov, Boundary properties of analytic functions (further contribution), Moscow Univ. Publ., M., 2012 (in Russian) | MR

[2] R. Nevanlinna, Odnoznachnye analiticheskie funktsii, GITTL, M.-L., 1941; R. Nevanlinna, Eindeutige analytische Funktionen, Springer-Verlag, Berlin, 1953 ; Springer-Verlag, Berlin, 1970 | MR | Zbl | Zbl

[3] I.I. Privalov, Boundary properties of analytic functions, GITTL, M. (in Russian)

[4] E.G. Rodikova, “On estimates for expansion coefficients for certain classes of functions analytic in a disk”, Proc. the VI Petrozavodsk International Conference “Complex Analysis and Applications”, Petrozavodsk, 2012, 64–69 (in Russian)

[5] E.G. Rodikova, “On coefficient multipliers in one weighted space of analytic in circle functions”, Vestnik Bryansk. Gosud. Univ., 4:2 (2012), 61–69 (in Russian)

[6] E.G. Rodikova, Factorization, characterization of root sets, and questions of interpolation in weighted spaces of analytic functions, PhD thesis, Bryansk, 2014 (in Russian)

[7] E.G. Rodikova, “Linear continuous functionals on Privalov spaces”, Proc. of 21th Inter. Saratov winter school “Modern problems of theory of functions and their applications”, Saratov, 2022, 249–251 (in Russian)

[8] E.G. Rodikova, “On linear continuous functionals on area Privalov classes”, Proc. of All-Russian scientific-practical conferennce “Theoretical and applied aspects of natural science education in digitalization age”, Bryansk, 2022, 102–103 (in Russian)

[9] E.G. Rodikova, “On coefficient multipliers for area Privalov classes”, Ufa Math. J., 13:4 (2021), 80–90 | DOI | MR | Zbl

[10] W. Rudin, Functional analysis, McGraw-Hill Book Comp., New York, 1973 | MR | MR | Zbl

[11] F.A. Shamoyan, “Parametric representation and description of the root sets of weighted classes of functions holomorphic in the disk”, Siberian Math. J., 40:6 (1999), 1211–1229 | DOI | MR | Zbl

[12] P. Duren, B. Romberg, A. Shields, “Linear functionals on $H^p$ spaces with $0

1$”, J. Reine Angew. Math., 238 (1969), 32–60 | MR | Zbl

[13] M. Pavlovic, Introduction to function spaces in a disk, Matematicki Institut SANU, Beograd, 2004 | MR

[14] E.G. Rodikova, “Coefficient multipliers for the Privalov class in a disk”, Zhurn. SFU. Ser. Matem. Fiz., 11:6 (2018), 723–732 | MR

[15] E.G. Rodikova, “Continuous linear functionals on the Nevanlinna-Djrbashian type spaces”, Int. Conf. «Complex Analysis and Related Topics». Abstracts, Proc. of the Math. Center named after N.I. Lobachevsky, 63, KFU, Kazan, 2022, 51–52 (in Russian)

[16] E.G. Rodikova, F.A. Shamoyan, “On the differentiation in the Privalov classes”, Zhurn. SFU. Ser. Matem. Fiz., 13:5 (2020), 622–630 | MR

[17] A.E. Taylor, “Banach spaces of functions analytic in the unit circle II”, Studia Math., 12 (1951), 25–50 | DOI | MR | Zbl

[18] N. Yanagihara, “Multipliers and linear functionals for the class $N^+$”, Transactions of the Amer. Math. Soc., 180 (1973), 449–461 | MR | Zbl

[19] K. Yoshida, Functional Analysis, Springer-Verlag, Berlin–Göttingen–Heidelberg, 1965 | Zbl