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A generalized Hilbert operator on Bloch space and BMOA spaces. (English) Zbl 1491.47029

Giri, Debasis (ed.) et al., Proceedings of the sixth international conference on mathematics and computing, ICMC 2020, Gangtok, Sikkim, India, March 18–20, 2020. Singapore: Springer. Adv. Intell. Syst. Comput. 1262, 441-449 (2021).
Summary: We consider the generalized Hilbert operators \(\mathcal{H}_{a,b}\), \[\mathcal{H}_{a,b}(f)(z)=\sum_{n=0}^{\infty }\left( \sum_{k=0}^{\infty }\frac{(b-a)_n\mu_{n,k}}{n!}a_k\right) z^n,\] where \(a,b\in \mathbb{C}\), \(f(z)=\sum_{k=0}^{\infty }a_kz^k\) analytic on the unit disc \(\mathbb{D}\), \(\mu\) be a positive Borel measure on the interval \([0, 1)\) and \(\mu_n\) denote the moment of order \(n\) of \(\mu \), that is, \( \mu_n=\int_{[0,1)}t^n\,\mathrm{d}\mu (t)\) with \(\mu_{n,k}=\mu_{n+k} \). This is one of the generalizations of the classical Hilbert operator. In this paper, we characterize the measures \(\mu\) and find the conditions on \(a, b\) such that \(\mathcal{H}_{a,b}\) is bounded and compact on Bloch space and BMOA spaces.
For the entire collection see [Zbl 1468.65003].

MSC:

47B91 Operators on complex function spaces
30H30 Bloch spaces
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