Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 39, 2014, 231-258

GENERALIZED HILBERT OPERATORS

Petros Galanopoulos, Daniel Girela, José Ángel Peláez and Aristomenis G. Siskakis

Aristotle University of Thessaloniki, Department of Mathematics
54124, Thessaloniki, Greece; galanopoulos_petros 'at' yahoo.gr

Universidad de Málaga, Departamento de Análisis Matemático
Campus de Teatinos, 29071 Málaga, Spain; girela 'at' uma.es

Universidad de Málaga, Departamento de Análisis Matemático
Campus de Teatinos, 29071 Málaga, Spain; japelaez 'at' uma.es

Aristotle University of Thessaloniki, Department of Mathematics
54124, Thessaloniki, Greece; siskakis 'at' math.auth.gr

Abstract. If g is an analytic function in the unit disc D, we consider the generalized Hilbert operator H_g defined by

H_g(f)(z) = \int_0^1 f(t)g'(tz) dt.

We study these operators acting on classical spaces of analytic functions in D. More precisely, we address the question of characterizing the functions g for which the operator H_g is bounded (compact) on the Hardy spaces H^p, on the weighted Bergman spaces A^p_\alpha or on the spaces of Dirichlet type D^p_\alpha.

2010 Mathematics Subject Classification: Primary 47B35; Secondary 30H10.

Key words: Generalized Hilbert operators, Hardy spaces, Dirichlet spaces, Bergman spaces.

Reference to this article: P. Galanopoulos, D. Girela, J.Á. Peláez and A.G. Siskakis: Generalized Hilbert operators. Ann. Acad. Sci. Fenn. Math. 39 (2014), 231-258.

Full document as PDF file

doi:10.5186/aasfm.2014.3912