A generalized Hilbert operator acting on conformally invariant spaces

Abstract

If $\mu$ is a positive Borel measure on the interval $[0,1)$ we let ${\mathcal{H}}_{\mu }$ be the Hankel matrix ${\mathcal{H}}_{\mu }=({\mu }_{n,k}{)}_{n,k\ge 0}$ with entries ${\mu }_{n,k}={\mu }_{n+k}$, where, for $n=0,1,2,\dots$, ${\mu }_n$ denotes the moment of orden $n$ of $\mu$. This matrix induces formally the operator $${\mathcal{H}}_{\mu }(f)(z)=\sum _{n=0}^{\infty }\left(\sum _{k=0}^{\infty }{\mu }_{n,k}{a}_k\right)z^n$$ on the space of all analytic functions $f(z)={\sum }_{k=0}^{\infty }{a}_k{z}^k$, in the unit disc $\D $. This is a natural generalization of the classical Hilbert operator. The action of the operators $H_{\mu }$ on Hardy spaces has been recently studied. This paper is devoted to study the operators $H_{\mu }$ acting on certain conformally invariant spaces of analytic functions on the disc such as the Bloch space, $BMOA$, the analytic Besov spaces, and the $Q_s$ spaces.

24 pages

Cited by in corpus (6)

• A Hankel matrix acting on spaces of analytic functions
• Mean Lipschitz spaces and a generalized Hilbert operator
• Generalized Hilbert Operator Acting on Weighted Bergman Spaces and on Dirichlet Spaces
• Generalized integral type Hilbert operator acting on weighted Bloch space
• Generalized Hilbert operators arising from Hausdorff matrices
• Hilbert-type operators acting between weighted Fock spaces