Products of Volterra type operator and composition operator from \(H^\infty \)and Bloch spaces to Zygmund spaces. (English) Zbl 1145.47022

Li, Songxiao; Stević, Stevo

J. Math. Anal. Appl. 345, No. 1, 40-52 (2008).

The Zygmund space \(\mathcal{Z}\) is the set of all analytic functions \(f\) on the unit disc \(\mathbb D\) such that \(\| f\| _{\mathcal{Z}}=| f(0)| +| f'(0)| +\sup_z(1-| z| ^2)| f''(z)| <+\infty,\) endowed with such a norm. If \(\lim_{| z| \to 1}(1-| z| ^2)| f''(z)| =0,\) then \(f\) is said to belong to the little Zygmund space \(\mathcal{Z}_0.\) Given an analytic function \(g\in H(\mathbb D),\) two types of Volterra integral operator are defined according to \(J_g(f)(z)=\int_0^z fg'\) and \(I_g(f)(z)=\int_0^z f'g\) for \(f\in H(\mathbb D).\) The authors consider \(\mathcal{Z}\) and \( \mathcal{Z}_0\) valued compositions of these operators and composition operators \(C_\varphi\) whose symbol \(\varphi\) is an analytic self-map of \(\mathbb D.\) They compare their boundedness or compactness regarding the domain space, specifically, \(H^\infty,\) the Bloch space \(\mathcal{B}, \) or the little Bloch space \(\mathcal{B}_0 .\) As a sample of the paper results, let us quote the following (Theorem 1): Set \(T=C_\varphi \circ I_g.\) Then \(T:H^\infty \to \mathcal{Z}\) is bounded if and only if \(T:\mathcal{B} \to \mathcal{Z}\) is bounded if and only if \(T:\mathcal{B}_0\to \mathcal{Z}\) is bounded. Similarly, for the compact case.

Reviewer: Pablo Galindo (Valencia)

MSC:

47B33	Linear composition operators
47B38	Linear operators on function spaces (general)

Keywords:

Volterra operator; composition operator; Zygmund space; Bloch space; bounded analytic functions

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, 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, 3, 423-432, (2003) · Zbl 1052.47044
[4]	Cowen, C.C.; MacCluer, B.D., Composition operators on spaces of analytic functions, (1995), CRC Press Boca Raton, FL · Zbl 0873.47017
[5]	Dunford, N.; Schwartz, J.T., Linear operators I, (1958), Interscience Publishers, Jon Willey and Sons New York
[6]	Duren, P.L., Theory of \(H^p\) spaces, (1970), Academic Press New York · Zbl 0215.20203
[7]	Hu, Z.J., Extended Cesàro operators on mixed norm spaces, Proc. amer. math. soc., 131, 7, 2171-2179, (2003) · Zbl 1054.47023
[8]	Hu, Z.J., Extended Cesàro operators on the Bloch space in the unit ball of \(\mathbb{C}^n\), Acta math. sci. ser. B engl. ed., 23, 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) · Zbl 1072.47029
[10]	Li, S., Riemann – stieltjes operators from \(F(p, q, s)\) to Bloch space on the unit ball, J. inequal. appl., 2006, (2006), Article ID 27874, 14 pp
[11]	Li, S.; Stević, S., Riemann – stieltjies type integral operators on the unit ball in \(\mathbb{C}^n\), Complex var. elliptic funct., 52, 6, 495-517, (2007) · Zbl 1124.47022
[12]	Li, S.; Stević, S., Volterra-type operators on Zygmund spaces, J. inequal. appl., 2007, (2007), Article ID 32124, 10 pp · Zbl 1146.30303
[13]	Madigan, K.; Matheson, A., Compact composition operators on the Bloch space, Trans. amer. math. soc., 347, 7, 2679-2687, (1995) · Zbl 0826.47023
[14]	Ohno, S., Weighted composition operators between \(H^\infty\) and the Bloch space, Taiwanese J. math., 5, 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
[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 \(\mathbb{C}^n\), J. inequal. appl., 1, 81-88, (2005) · Zbl 1074.47013
[18]	Stević, S., Boundedness and compactness of an integral operator on mixed norm spaces on the polydisc, Sibirsk. mat. zh., 48, 3, 694-706, (2007) · Zbl 1164.47331
[19]	Tang, X., Extended Cesàro operators between Bloch-type spaces in the unit ball of \(\mathbb{C}^n\), J. math. anal. appl., 326, 1199-1211, (2007) · Zbl 1117.47022
[20]	Zhu, K., Bloch type spaces of analytic functions, Rocky mountain J. math., 23, 3, 1143-1177, (1993) · Zbl 0787.30019