A Hankel matrix acting on spaces of analytic functions. (English) Zbl 1508.47056
Summary: If \(\mu \) is a positive Borel measure on the interval \([0,1)\) we let \(\mathcal H_\mu \) be the Hankel matrix \(\mathcal H_\mu =(\mu_{n, k})_{n,k\geq 0}\) 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 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 \(f(z)=\sum_{k=0}^\infty a_kz^k\) in the unit disc \(\mathbb D\). This is a natural generalization of the classical Hilbert operator. In this paper we improve the results obtained in some recent papers concerning the action of the operators \(\mathcal {H}_\mu \) on Hardy spaces and on Möbius invariant spaces.
MSC:
| 47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |
| 30H05 | Spaces of bounded analytic functions of one complex variable |
| 30H10 | Hardy spaces |
Keywords:
Hankel matrix; generalized Hilbert operator; Hardy spaces; BMOA; Bloch space; conformally invariant spaces; Carleson measuresReferences:
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