Wang, Maofa; Wang, Shuming Weighted composition operators between \(n\)-th \(\alpha\)-weighted spaces. (English) Zbl 1523.47040 Rocky Mt. J. Math. 51, No. 4, 1503-1519 (2021). Summary: Let \(V_n^\alpha\) be the Banach space of holomorphic functions on the open unit disk \(\mathbb{D}\) in the complex plane consisting of those \(f\) such that \(\parallel f\parallel_{V_{n}^{\alpha}}: = \sum_{i = 0}^{n - 1} | f^{(i)}(0) | + \sup_{z \in\mathbb{D}} (1 - |z|^2)^\alpha | f^{(n)} (z) | < \infty\) and \(V_{n,0}^\alpha\) be the closed subspace of \(V_n^\alpha\) consisting of those \(f\) for which \(\lim_{|z| \to 1}(1 - |z|^2)^\alpha | f^{(n)}(z)| = 0\), where \(n\) is any nonnegative integer and \(\alpha > 0\). We give boundedness characterizations, norm estimates and essential norm estimates of weighted composition operators \(W_{\psi, \varphi} : V_n^\alpha \to V_m^\beta\) and \(W_{\psi, \varphi} : V_{n,0}^\alpha \to V_{m, 0}^\beta\), respectively, where \(W_{\psi, \varphi} f(z) = \psi (z) f (\varphi(z))\). As a corollary, we characterize the compactness of \(W_{\psi, \varphi}\). Specifically, our characterizations involve not only the classical Julia-Carathéodory type condition, but also the powers \(\varphi^k\). In addition, our results extend several well-known results in the literature. MSC: 47B33 Linear composition operators 30H30 Bloch spaces 46E15 Banach spaces of continuous, differentiable or analytic functions Keywords:weighted composition operator; iterated weighted-type space; Bloch-type space; Zygmund-type space; essential norm PDFBibTeX XMLCite \textit{M. Wang} and \textit{S. Wang}, Rocky Mt. J. Math. 51, No. 4, 1503--1519 (2021; Zbl 1523.47040) References: [1] F. Colonna and N. Hmidouch, “Weighted composition operators on iterated weighted-type Banach spaces of analytic functions”, Complex Anal. Oper. Theory 13:4 (2019), 1989-2016. · Zbl 1436.47006 [2] C. C. Cowen and B. D. MacCluer, Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. · Zbl 0873.47017 [3] K. Esmaeili and M. Lindström, “Weighted composition operators between Zygmund type spaces and their essential norms”, Integral Equations Operator Theory 75:4 (2013), 473-490. · Zbl 1306.47036 [4] O. Hyvärinen and M. Lindström, “Estimates of essential norms of weighted composition operators between Bloch-type spaces”, J. Math. Anal. Appl. 393:1 (2012), 38-44. · Zbl 1267.47040 [5] O. Hyvärinen, M. Kemppainen, M. Lindström, A. Rautio, and E. Saukko, “The essential norm of weighted composition operators on weighted Banach spaces of analytic functions”, Integral Equations Operator Theory 72:2 (2012), 151-157. · Zbl 1252.47026 [6] B. D. MacCluer and R. Zhao, “Essential norms of weighted composition operators between Bloch-type spaces”, Rocky Mountain J. Math. 33:4 (2003), 1437-1458. · Zbl 1061.30023 [7] J. S. Manhas and R. Zhao, “New estimates of essential norms of weighted composition operators between Bloch type spaces”, J. Math. Anal. Appl. 389:1 (2012), 32-47. · Zbl 1267.47042 [8] A. Montes-Rodríguez, “Weighted composition operators on weighted Banach spaces of analytic functions”, J. London Math. Soc. \[(2) 61\]:3 (2000), 872-884. · Zbl 0959.47016 [9] S. Ohno, K. Stroethoff, and R. Zhao, “Weighted composition operators between Bloch-type spaces”, Rocky Mountain J. Math. 33:1 (2003), 191-215. · Zbl 1042.47018 [10] K. H. Zhu, “Bloch type spaces of analytic functions”, Rocky Mountain J. Math. 23:3 (1993), 1143-1177 · 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.