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Products of Volterra type operator and composition operator from ${H}^{\infty }$and Bloch spaces to Zygmund spaces. (English) Zbl 1145.47022
The Zygmund space $𝒵$ is the set of all analytic functions $f$ on the unit disc $𝔻$ such that ${\parallel f\parallel }_{𝒵}=|f\left(0\right)|+|{f}^{\text{'}}\left(0\right)|+{sup}_{z}{\left(1-|z|}^{2}\right)|{f}^{\text{'}\text{'}}\left(z\right)|<+\infty ,$ endowed with such a norm. If ${lim}_{|z|\to 1}{\left(1-|z|}^{2}\right)|{f}^{\text{'}\text{'}}\left(z\right)|=0,$ then $f$ is said to belong to the little Zygmund space ${𝒵}_{0}·$ Given an analytic function $g\in H\left(𝔻\right),$ two types of Volterra integral operator are defined according to ${J}_{g}\left(f\right)\left(z\right)={\int }_{0}^{z}f{g}^{\text{'}}$ and ${I}_{g}\left(f\right)\left(z\right)={\int }_{0}^{z}{f}^{\text{'}}g$ for $f\in H\left(𝔻\right)·$ The authors consider $𝒵$ and ${𝒵}_{0}$ valued compositions of these operators and composition operators ${C}_{\phi }$ whose symbol $\phi$ is an analytic self-map of $𝔻·$ They compare their boundedness or compactness regarding the domain space, specifically, ${H}^{\infty },$ the Bloch space $ℬ,$ or the little Bloch space ${ℬ}_{0}·$ As a sample of the paper results, let us quote the following (Theorem 1): Set $T={C}_{\phi }\circ {I}_{g}·$ Then $T:{H}^{\infty }\to 𝒵$ is bounded if and only if $T:ℬ\to 𝒵$ is bounded if and only if $T:{ℬ}_{0}\to 𝒵$ is bounded. Similarly, for the compact case.

MSC:
 47B33 Composition operators 47B38 Operators on function spaces (general)
References:
 [1] Benke, G.; Chang, D. C.: A note on weighted Bergman spaces and the Cesàro operator, Nagoya math. J. 159, 25-43 (2000) · Zbl 0981.32001 [2] Chang, D. C.; Li, S.; Stević, S.: On some integral operators on the unit polydisk and the unit ball, Taiwanese J. Math. 11, No. 5, 1251-1286 (2007) · Zbl 1149.47026 [3] Chang, D. C.; Stević, S.: Estimates of an integral operator on function spaces, Taiwanese J. Math. 7, No. 3, 423-432 (2003) · Zbl 1052.47044 [4] Cowen, C. C.; Maccluer, B. D.: Composition operators on spaces of analytic functions, (1995) · Zbl 0873.47017 [5] Dunford, N.; Schwartz, J. T.: Linear operators I, (1958) · Zbl 0084.10402 [6] Duren, P. L.: Theory of hp spaces, (1970) · Zbl 0215.20203 [7] Hu, Z. J.: Extended Cesàro operators on mixed norm spaces, Proc. amer. Math. soc. 131, No. 7, 2171-2179 (2003) · Zbl 1054.47023 · doi:10.1090/S0002-9939-02-06777-1 [8] Hu, Z. J.: Extended Cesàro operators on the Bloch space in the unit ball of cn, Acta math. Sci. ser. B engl. Ed. 23, No. 4, 561-566 (2003) · Zbl 1044.47023 [9] Hu, Z. J.: Extended Cesàro operators on Bergman spaces, J. math. Anal. appl. 296, 435-454 (2004) [10] Li, S.: Riemann – Stieltjes operators from $F\left(p,q,s\right)$ to Bloch space on the unit ball, J. inequal. Appl. 2006 (2006) · Zbl 1131.47030 · doi:10.1155/JIA/2006/27874 [11] Li, S.; Stević, S.: Riemann – stieltjies type integral operators on the unit ball in cn, Complex var. Elliptic funct. 52, No. 6, 495-517 (2007) · Zbl 1124.47022 · doi:10.1080/17476930701235225 [12] Li, S.; Stević, S.: Volterra-type operators on Zygmund spaces, J. inequal. Appl. 2007 (2007) · Zbl 1146.30303 · doi:10.1155/2007/32124 [13] Madigan, K.; Matheson, A.: Compact composition operators on the Bloch space, Trans. amer. Math. soc. 347, No. 7, 2679-2687 (1995) · Zbl 0826.47023 · doi:10.2307/2154848 [14] Ohno, S.: Weighted composition operators between H$\infty$ and the Bloch space, Taiwanese J. Math. 5, No. 2, 555-563 (2001) · Zbl 0997.47025 [15] Pommerenke, Ch.: Schlichte funktionen und analytische funktionen von beschränkter mittlerer oszillation, Comment. math. Helv. 52, 591-602 (1977) · Zbl 0369.30012 · doi:10.1007/BF02567392 [16] Siskakis, A. G.; Zhao, R.: A Volterra type operator on spaces of analytic functions, Contemp. math. 232, 299-311 (1999) · Zbl 0955.47029 [17] Stević, S.: On an integral operator on the unit ball in cn, J. inequal. Appl. 1, 81-88 (2005) · Zbl 1074.47013 · doi:10.1155/JIA.2005.81 [18] Stević, S.: Boundedness and compactness of an integral operator on mixed norm spaces on the polydisc, Sibirsk. mat. Zh. 48, No. 3, 694-706 (2007) · Zbl 1164.47331 · doi:emis:journals/SMZ/2007/03/694.htm [19] Tang, X.: Extended Cesàro operators between Bloch-type spaces in the unit ball of cn, J. math. Anal. appl. 326, 1199-1211 (2007) · Zbl 1117.47022 · doi:10.1016/j.jmaa.2006.03.082 [20] Zhu, K.: Bloch type spaces of analytic functions, Rocky mountain J. Math. 23, No. 3, 1143-1177 (1993) · Zbl 0787.30019 · doi:10.1216/rmjm/1181072549