## Banach Journal of Mathematical Analysis

### 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 $\mathcal{H}_{\mu}=(\mu_{n,k})_{n,k\ge0}$ with entries $\mu_{n,k}=\mu_{n+k}$, where, for $n=0,1,2,\dots$, $\mu_{n}$ denotes the moment of order $n$ of $\mu$. This matrix formally induces the operator

$$\mathcal{H}_{\mu}(f)(z)=\sum_{n=0}^{\infty}(\sum_{k=0}^{\infty}\mu_{n,k}{a_{k}})z^{n}$$ on the space of all analytic functions $f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$, in the unit disk $\mathbb{D}$. 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 article is devoted to a study of the operators $H_{\mu}$ acting on certain conformally invariant spaces of analytic functions on the disk such as the Bloch space, the space BMOA, the analytic Besov spaces, and the $Q_{s}$-spaces.

#### Article information

Source
Banach J. Math. Anal., Volume 12, Number 2 (2018), 374-398.

Dates
Accepted: 16 May 2017
First available in Project Euclid: 8 September 2017

https://projecteuclid.org/euclid.bjma/1504857614

Digital Object Identifier
doi:10.1215/17358787-2017-0023

Mathematical Reviews number (MathSciNet)
MR3779719

Zentralblatt MATH identifier
06873506

#### Citation

Girela, Daniel; Merchán, Noel. A generalized Hilbert operator acting on conformally invariant spaces. Banach J. Math. Anal. 12 (2018), no. 2, 374--398. doi:10.1215/17358787-2017-0023. https://projecteuclid.org/euclid.bjma/1504857614

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