Annales Academić Scientiarum Fennicć
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 H^p_\alpha (\alpha \ge 0), Bergman spaces with logarithmic weights, etc. In particular, we extend Diamantopoulos-Siskakis result by proving that H maps H^p_\alpha into H^p_\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.

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doi:10.5186/aasfm.2012.3715