Colonna, Flavia; Hmidouch, Nacir Weighted composition operators between weighted-type spaces and the Bloch space and BMOA. (English) Zbl 1513.47047 Adv. Oper. Theory 5, No. 1, 94-114 (2020). Summary: Let \(\mathbb{D}\) denote the open unit disk in \(\mathbb{C} \). For an integer \(n\ge 0\), let \(V_n\) be the space defined recursively by \[ V_0=\big \{f:\mathbb{D}\rightarrow\mathbb{C}: f \text{ analytic, } |f(z)|=O\big ((1-|z|)^{-1}\big)\big \}, \] and for \(n\ge 1\), \(f\in V_n\) if and only if \(f'\in V_{n-1} \). In this work, we characterize the bounded and the compact weighted composition operators between \(V_n\) and the Bloch space and the space BMOA of analytic functions of bounded mean oscillation, respectively. We also show that the bounded (respectively, compact) weighted composition operators mapping the space BMOA into \(V_n\) are precisely the same as the bounded (respectively, compact) weighted composition operators mapping the Bloch space into \(V_n\). Cited in 1 Document MSC: 47B33 Linear composition operators 30H30 Bloch spaces 30H10 Hardy spaces Keywords:Bloch space; BMOA; Zygmund space; weighted composition operator PDFBibTeX XMLCite \textit{F. Colonna} and \textit{N. Hmidouch}, Adv. Oper. Theory 5, No. 1, 94--114 (2020; Zbl 1513.47047) Full Text: DOI References: [1] Colonna, F.; Hmidouch, N., Weighted composition operators on iterated weighted-type Banach spaces of analytic functions, Complex Anal. Oper. Theory, 13, 4, 1989-2016 (2019) · Zbl 1436.47006 [2] Colonna, F.; Tjani, M., Operator norms and essential norms of weighted composition operators between Banach spaces of analytic functions, J. Math. Anal. Appl., 434, 1, 93-124 (2016) · Zbl 1338.30049 [3] Conway, Jb, Functions of One Complex Variable I (2001), New York: Springer, New York [4] Duren, Pl; Romberg, Bw; Shields, Al, Linear functionals on \(H^p\) spaces with \(0<p<1\), J. Reine Angew. Math., 238, 32-60 (1969) · Zbl 0176.43102 [5] Girela, D.: Analytic functions of bounded mean oscillation, Complex Function Spaces (Mekrijärvi, 1999), 61-170, Univ. Joensuu Dept. Math. Rep. Ser., 4, Univ. Joensuu, Joensuu (2001) · Zbl 0981.30026 [6] Hmidouch, N.: Weighted composition operators acting on some classes of Banach spaces of analytic functions, Doctoral Dissertation, George Mason University (2017) [7] Laitila, J., Weighted composition operators on BMOA, Comput. Methods Funct. Theory, 9, 1, 27-46 (2009) · Zbl 1163.47018 [8] Li, S.; Stević, S., Weighted composition operators from Zygmund spaces into Bloch spaces, Appl. Math. Comput., 206, 825-831 (2008) · Zbl 1215.47022 [9] Ohno, S.; Stroethoff, K.; Zhao, R., Weighted composition operators between Bloch-type spaces, Rocky Mountain J. Math., 33, 1, 191-215 (2003) · Zbl 1042.47018 [10] Ohno, S.; Zhao, R., Weighted composition operators on the Bloch space, Bull. Austral. Math. Soc., 63, 2, 177-185 (2001) · Zbl 0985.47022 [11] Stević, S., Weighted differentiation composition operators from \(H^{\infty }\) and Bloch spaces to \(n\) th weighted-type spaces on the unit disk, Appl. Math. Comput., 216, 3634-3641 (2010) · Zbl 1195.30073 [12] Tjani, M.: Compact composition operators on some Möbius invariant Banach spaces, Doctoral Dissertation, Michigan State University (1996) [13] Ye, S.; Hu, Q., Weighted composition operators on the Zygmund space, Abs. Appl. Anal., 462482, 18 (2012) · Zbl 1277.47038 [14] Zhu, K., Operator Theory in Function Spaces (1990), New York: Marcel Dekker, New York [15] Zhu, K., Bloch type spaces of analytic functions, Rocky Mountain. 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.