Ye, Shanli; Zhou, Zhihui A derivative-Hilbert operator acting on the Bloch space. (English) Zbl 1494.47054 Complex Anal. Oper. Theory 15, No. 5, Paper No. 88, 16 p. (2021). Summary: Let \(\mu\) be a positive Borel measure on the interval [0,1). Suppose \(\mathcal{H}_\mu\) is the Hankel matrix \((\mu_{n,k})_{n,k\ge 0}\) with entries \(\mu_{n,k}=\mu_{n+k} \), where \(\mu_n=\int_{[0,1)}t^n\,d\mu (t)\). The matrix formally induces the operator \(\mathcal{H}_\mu (f)(z)=\sum_{n=0}^{\infty}\big (\sum_{k=0}^{\infty }\mu_{n,k}a_k\big )z^n,\) which has been widely studied in [G.-L. Bao and H. Wulan, J. Math. Anal. Appl. 409, No. 1, 228–235 (2014; Zbl 1326.47028); C. Chatzifountas et al., ibid. 413, No. 1, 154–168 (2014; Zbl 1308.42021); P. Galanopoulos and J. Á. Peláez, Stud. Math. 200, No. 3, 201–220 (2010; Zbl 1206.47024); D. Girela and N. Merchán, Banach J. Math. Anal. 12, No. 2, 374–398 (2018; Zbl 1496.47054)]. In this paper, we define the derivative-Hilbert operator as \[ \mathcal{DH}_{\mu}(f)(z)=\sum_{n=0}^{\infty} \left( \sum_{k=0}^{\infty} \mu_{n, k} a_k\right) (n+1)z^n. \] We mainly characterize the measures \(\mu\) for which \(\mathcal{DH}_{\mu}\) is a bounded (resp., compact) operator on the Bloch space \(\mathscr{B} \). We also characterize those measures \(\mu\) for which \(\mathcal{DH}_{\mu}\) is a bounded (resp., compact) operator from the Bloch space \(\mathscr{B}\) into the Bergman space \(A^p\), \(1\le p<\infty \). Cited in 1 ReviewCited in 8 Documents MSC: 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators 30H30 Bloch spaces 30H20 Bergman spaces and Fock spaces Keywords:Hilbert operator; Bloch space; Bergman space Citations:Zbl 1326.47028; Zbl 1308.42021; Zbl 1206.47024; Zbl 1496.47054 PDFBibTeX XMLCite \textit{S. Ye} and \textit{Z. Zhou}, Complex Anal. Oper. Theory 15, No. 5, Paper No. 88, 16 p. (2021; Zbl 1494.47054) Full Text: DOI References: [1] Aleman, A.; Montes-Rodríguez, A.; Sarafoleanu, A., The eigenfunctions of the Hilbert matrix, Const. Approx., 36, 353-374 (2012) · Zbl 1268.47040 [2] Bao, G.; Wulan, H., Hankel matrices acting on Dirichlet spaces, J. Math. Anal. Appl., 409, 228-235 (2014) · Zbl 1326.47028 [3] Chatzifountas, Ch; Girela, D.; Peláez, JÁ, A generalized Hilbert matrix acting on Hardy spaces, J. Math. Anal. Appl., 413, 154-168 (2014) · Zbl 1308.42021 [4] Cowen, C. C., MacCluer, B. D.: Composition Operators on Spaces of Analytic Functions. Studies in Advanced Mathematics,CRC Press, Boca Raton, Fla, USA (1995) · Zbl 0873.47017 [5] Diamantopoulos, E.; Siskakis, AG, Composition operators and the Hilbert matrix, Studia Math., 140, 191-198 (2000) · Zbl 0980.47029 [6] Duren, PL, Theory of \({H}^p\) Spaces (1970), New York: Academic Press, New York · Zbl 0215.20203 [7] Duren, PL; Schuster, A., Bergman Spaces. Mathematical Surveys and Monographs (2004), Providence, R.I.: American Mathematical Society, Providence, R.I. · Zbl 1059.30001 [8] Galanopoulos, P.; Girela, D.; Peláez, JÁ; Siskakis, AG, Generalized Hilbert operators, Ann. Acad. Sci. Fenn. Math., 39, 231-258 (2014) · Zbl 1297.47030 [9] Galanopoulos, P.; Peláez, JÁ, A Hankel matrix acting on Hardy and Bergman spaces, Studia Math., 200, 201-220 (2010) · Zbl 1206.47024 [10] Girela, D.; Merchán, N., A generalized Hilbert operator acting on conformally invariant spaces, Banach J. Math. Anal., 12, 374-398 (2018) · Zbl 1496.47054 [11] Gnuschke-Hauschild, D.; Pommerenke, C., On Bloch functions and gap series, J. Reine. Angew. Math., 367, 172-186 (1986) · Zbl 0576.30027 [12] Hastings, WW, A Carleson measure theorem for Bergman spaces, Studia Math., 52, 237-241 (1975) · Zbl 0296.31009 [13] Maccluer, B.; Zhao, R., Vanishing logarithmic Carleson measures, Illinois J. Math., 46, 507-518 (2002) · Zbl 1020.30038 [14] Pommerenke, Ch; Clunie, J.; Anderson, J., On Bloch functions and normal functions, J. Reine. Angew. Math., 270, 12-37 (1974) · Zbl 0292.30030 [15] Zhao, R., On logarithmic Carleson measures, Acta Sci. Math., 69, 605-618 (2003) · Zbl 1050.30024 [16] Zhu, K., Bloch Type Spaces of Analytic Functions, Rocky MT. J. Math., 23, 1143-1177 (1993) · Zbl 0787.30019 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.