Generalized Hilbert Operators

Abstract

If $g$ is an analytic function in the unit disc $\D $ we consider the generalized Hilbert operator $\hg$ defined by {equation*}\label{H-g} \mathcal{H}_g(f)(z)=\int_0^1f(t)g'(tz)\,dt. {equation*} 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 $\hg $ is bounded (compact) on the Hardy spaces $H^p$, on the weighted Bergman spaces $A_{\alpha }^p$ or on the spaces of Dirichlet type ${\mathcal{D}}_{\alpha }^p$.

Cited by in corpus (7)

• A Hankel matrix acting on spaces of analytic functions
• Generalized Hilbert Operator Acting on Weighted Bergman Spaces and on Dirichlet Spaces
• Superposition operators, Hardy spaces, and Dirichlet type spaces
• Hankel matrices acting on the Hardy space $H^1$ and on Dirichlet spaces
• Generalized Hilbert matrix operators acting on Bergman spaces
• Hilbert-type operator induced by radial weight
• Hilbert-type operators acting between weighted Fock spaces