Weighted composition operators from Bergman-type spaces into Bloch spaces. (English) Zbl 1130.47016

It is a quite common trend in the study of composition operators between spaces of analytic functions in the unit disc of the complex plane that if boundedness corresponds to a “big-oh” condition, then compactness is determined by the “little-oh” of such condition; see, for instance, Chapter 3 in [J.H.Shapiro, “Composition operators and classical function theory” (Universitext: Tracts in Mathematics; New York: Springer–Verlag (1993; Zbl 0791.30033)]. The present article follows this trend. Given \(0<p<+\infty\) and a continuous function \(\phi:[0,1)\to (0,+\infty)\) such that for some \(0<s<t,\) \(\phi(r)=o(1-r)^s\) and \((1-r)^t=o(\phi(r))\) as \(r\to 1,\) the weighted Bergman-type space \(H(p,p,\phi)\) is the space of all analytic functions \(f\) in the unit disc such that \(\| f\| _{p,\phi}:=\int_{\mathbb D}| f(z)| ^p {\phi^p(| z| )\over{1-| z| }}\, dA(z)\) is finite, where \(dA\) is the normalized Lebesgue measure in the unit disc. The authors prove that a weighted composition operator \(uC_\varphi\) acting from \(H(p,p,\phi)\) into the Bloch space \({\mathcal B}\) is bounded if and only if \[ (1-| z| ^2)| u'(z)| =O\left(\phi(| \varphi(z)| )(1-| \varphi(z)| ^2)^{1/ p}\right) \text{ and } \]
\[ \;(1-| z| ^2)| u(z)\varphi'(z)| =O\left(\phi(| \varphi(z)| )(1-| \varphi(z)| ^2)^{1+1/ p}\right). \] The “little-oh” of such condition when \(| \varphi(z)| \to 1,\) respectively when \(| z| \to 1,\) characterizes the compactness of \(uC_\varphi: H(p,p,\phi)\to {\mathcal B},\) respectively \(uC_\varphi: H(p,p,\phi)\to {\mathcal B}_0,\) the little Bloch space. The subsequent corollaries for the Bergman spaces \(A^p=H(p,p,(1-r)^{1/p})\) are stated.


47B33 Linear composition operators
46E15 Banach spaces of continuous, differentiable or analytic functions
30H05 Spaces of bounded analytic functions of one complex variable
30D45 Normal functions of one complex variable, normal families


Zbl 0791.30033
Full Text: DOI arXiv


[1] Cowen C C and MacCluer B D, Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics (Boca Raton: CRC Press) (1995) · Zbl 0873.47017
[2] Hu Z J, Extended Cesàro operators on mixed norm spaces, Proc. Am. Math. Soc. 131(7) (2003) 2171–2179 · Zbl 1054.47023 · doi:10.1090/S0002-9939-02-06777-1
[3] MacCluer B D and Zhao R, Essential norms of weighted composition operators between Bloch-type spaces, Rocky Mountain J. Math. 33(4) (2003) 1437–1458 · Zbl 1061.30023 · doi:10.1216/rmjm/1181075473
[4] Madigan K and Matheson A, Compact composition operators on the Bloch space, Trans. Am. Math. Soc. 347(7) (1995) 2679–2687 · Zbl 0826.47023 · doi:10.2307/2154848
[5] Ohno S, Weighted composition operators between H and the Bloch space, Taiwan. J. Math. 5(3) (2001) 555–563 · Zbl 0997.47025
[6] Ohno S, Stroethoff K and Zhao R, Weighted composition operators between Bloch-type spaces, Rocky Mountain J. Math. 33 (2003) 191–215 · Zbl 1042.47018 · doi:10.1216/rmjm/1181069993
[7] Ohno S and Zhao R, Weighted composition operators on the Bloch space, Bull. Austral. Math. Soc. 63 (2001) 177–185 · Zbl 0985.47022 · doi:10.1017/S0004972700019250
[8] Rudin W, Function theory in the unit ball of \(\mathbb{C}\)n (New York: Springer Verlag) (1980) · Zbl 0495.32001
[9] Shi J H and Luo L, Composition operators on the Bloch space, Acta Math. Sinica 16 (2000) 85–98 · Zbl 0967.32007 · doi:10.1007/s101149900028
[10] Shields A L and Williams D L, Bounded projections, duality, and multipliers in spaces of analytic functions, Trans. Am. Math. Soc. 162 (1971) 287–302 · Zbl 0227.46034
[11] Stević S, On an integral operator on the unit ball in \(\mathbb{C}\)n, J. Inequal. Appl. 1 (2005) 81–88 · Zbl 1074.47013 · doi:10.1155/JIA.2005.81
[12] Zhou Z H and Shi J H, Composition operators on the Bloch space in polydiscs, Complex Variables 46(1) (2001) 73–88 · Zbl 1026.47018
[13] Zhou Z, Composition operators between p-Bloch space and q-Bloch space in the unit ball, Progress in Natural Sci. 13(3) (2003) 233–236 · Zbl 1039.32006
[14] Zhu K, Operator Theory on Function Spaces, Marcel Dekker Inc. Pure and Applied Mathematics 139 (New York and Basel) (1990)
[15] Zhu K, Spaces of Holomorphic Functions in the Unit Ball (New York: Springer) (2005) · Zbl 1067.32005
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.