Mean Lipschitz spaces and a generalized Hilbert operator

• Autores: Noel Merchán
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 70, Fasc. 1, 2019, págs. 59-69
• Idioma: inglés
• DOI: 10.1007/s13348-018-0217-y
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• Resumen

  • 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 order n of \mu. This matrix induces formally the operator \begin{aligned}\mathcal {H}_\mu (f)(z)= \sum _{n=0}^{\infty }\left( \sum _{k=0}^{\infty } \mu _{n,k}{a_k}\right) z^n\end{aligned} 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 study the action of the operators \mathcal {H}_\mu on mean Lipschitz spaces of analytic functions.