Annales Academię Scientiarum Fennicę
Mathematica
Volumen 37, 2012, 161-174

HILBERT MATRIX OPERATOR ON SPACES OF ANALYTIC FUNCTIONS

Bartosz Lanucha, Maria Nowak and Miroslav Pavlovic

Uniwersytet Marii Curie-Sklodowskiej w Lublinie, Instytut Matematyki
pl. M. Curie-Sklodowskiej 1, 20-031 Lublin, Poland; bartosz.lanucha 'at' gmail.com

Uniwersytet Marii Curie-Sklodowskiej w Lublinie, Instytut Matematyki
pl. M. Curie-Sklodowskiej 1, 20-031 Lublin, Poland; mt.nowak 'at' poczta.umcs.lublin.pl

Univerzitet u Beogradu, Matematicki Fakultet
Studentski trg 16, 11001 Beograd, p.p. 550, Serbia; pavlovic 'at' matf.bg.ac.rs

Abstract. We consider the action of the Hilbert matrix operator, H, on the Hardy space H1, weighted Hardy spaces Hp\alpha (\alpha \ge 0), Bergman spaces with logarithmic weights, etc. In particular, we extend Diamantopoulos-Siskakis result by proving that H maps Hp\alpha into Hp\alpha if and only if \alpha +1/p < 1. A criterion for Hf to belong to H1 is given provided the coefficients of f are nonnegative. Also, H maps the A2-space with weight log\alpha(2/(1 - |z|2)) into the ordinary Bergman space A2 if \alpha > 3. Similarly, the Bloch space with logarithmic weight is mapped by H into the ordinary Bloch space.

2010 Mathematics Subject Classification: Primary 47B37, 30H10, 30H20, 30H30.

Key words: Hilbert matrix, Hardy spaces, Bergman spaces, Bloch and Besov spaces.

Reference to this article: B. Lanucha, M. Nowak and M. Pavlovic: Hilbert matrix operator on spaces of analytic functions. Ann. Acad. Sci. Fenn. Math. 37 (2012), 161-174.

Full document as PDF file

doi:10.5186/aasfm.2012.3715

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