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Girela, Daniel; Merchán, Noel

A generalized Hilbert operator acting on conformally invariant spaces. (English) Zbl 1496.47054

Banach J. Math. Anal. 12, No. 2, 374-398 (2018).
Summary: If \(\mu\) is a positive Borel measure on the interval \([0,1)\), we let \(\mathcal{H}_{\mu}\) be the Hankel matrix \(\mathcal{H}_{\mu}=(\mu_{n,k})_{n,k\geq0}\) 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 (cf. [C. Chatzifountas et al., J. Math. Anal. Appl. 413, No. 1, 154–168 (2014; Zbl 1308.42021)]). 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.

Cited in 1 Review
Cited in 22 Documents

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
30H10 Hardy spaces

Keywords:

Hilbert operators; conformally invariant spaces; Carleson measures

Citations:

Zbl 1308.42021
PDFBibTeX XMLCite
\textit{D. Girela} and \textit{N. Merchán}, Banach J. Math. Anal. 12, No. 2, 374--398 (2018; Zbl 1496.47054)
Full Text: DOI arXiv Euclid

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