On a new operator from the logarithmic Bloch space to the Bloch-type space on the unit ball. (English) Zbl 1162.47029

Stević, Stevo

Appl. Math. Comput. 206, No. 1, 313-320 (2008).

Summary: We study the boundedness and compactness of a recently introduced operator denoted by \(P^g_{\varphi}\) [cf. Discrete Dyn. Nat. Soc. 2008, Article ID 154263 (2008; Zbl 1155.32002)], which is a kind of the product of composition and integral operators on the unit ball \(\mathbb B \subset \mathbb C^n\), from the logarithmic Bloch space \(\mathcal B_{\log}\) and the little logarithmic Bloch space \(\mathcal B_{\log,0}\) to the Bloch-type space \(\mathcal B_{\mu}\) or the little Bloch-type space \(\mathcal B_{\mu,0}\).

MSC:

47B38	Linear operators on function spaces (general)
46E15	Banach spaces of continuous, differentiable or analytic functions
30H05	Spaces of bounded analytic functions of one complex variable

Keywords:

product of integral and composition operator; boundedness; compactness; weighted-type space; logarithmic Bloch space

Citations:

Zbl 1155.32002

References:

[1]	Avetisyan, K.L., Hardy-Bloch type spaces and lacunary series on the polydisk, Glasgow math. J., 49, 2, 345-356, (2007) · Zbl 1123.32004
[2]	Brown, L.; Shields, A.L., Multipliers and cyclic vectors in the Bloch space, Michigan math. J., 38, 141-146, (1991) · Zbl 0749.30023
[3]	Chang, D.C.; Li, S.; Stević, S., On some integral operators on the unit polydisk and the unit ball, Taiwan J. math., 11, 5, 1251-1286, (2007) · Zbl 1149.47026
[4]	Chang, D.C.; Stević, S., Estimates of an integral operator on function spaces, Taiwanese J. math., 7, 3, 423-432, (2003) · Zbl 1052.47044
[5]	Chang, D.C.; Stević, S., The generalized Cesàro operator on the unit polydisk, Taiwanese J. math., 7, 2, 293-308, (2003) · Zbl 1065.47033
[6]	Chang, D.C.; Stević, S., Addendum to the paper “A note on weighted Bergman spaces and the Cesàro operator”, Nagoya math. J., 180, 77-90, (2005) · Zbl 1090.32500
[7]	D. Clahane, S. Stević, Norm equivalence and composition operators between Bloch/Lipschitz spaces of the unit ball, J. Inequal. Appl., Article ID 61018, 2006, 11 pp. · Zbl 1131.47018
[8]	Cowen, C.C.; MacCluer, B.D., Composition operators on spaces of analytic functions, (1995), CRC Press Boca Raton, FL · Zbl 0873.47017
[9]	Hu, Z., Extended Cesàro operators on mixed norm spaces, Proc. amer. math. soc., 131, 7, 2171-2179, (2003) · Zbl 1054.47023
[10]	Hu, Z., 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
[11]	Hu, Z., Extended Cesàro operators on Bergman spaces, J. math. anal. appl., 296, 435-454, (2004) · Zbl 1072.47029
[12]	Li, S.; Stević, S., Integral type operators from mixed-norm spaces to \(\alpha\)-Bloch spaces, Integral transform. spec. funct., 18, 7, 485-493, (2007) · Zbl 1131.47031
[13]	Li, S.; Stević, S., Riemann – stieltjes operators on Hardy spaces in the unit ball of \(\mathbb{C}^n\), Bull. belg. math. soc. Simon stevin, 14, 621-628, (2007) · Zbl 1136.47023
[14]	Li, S.; Stević, S., Riemann – stieltjes type integral operators on the unit ball in \(\mathbb{C}^n\), Complex variables elliptic funct., 52, 6, 495-517, (2007) · Zbl 1124.47022
[15]	Li, S.; Stević, S., Weighted composition operators from \(\alpha\)-Bloch space to \(H^\infty\) on the polydisc, Numer. funct. anal. optimization, 28, 7-8, 911-925, (2007) · Zbl 1130.47015
[16]	S. Li, S. Stević, Weighted composition operators from \(H^\infty\) to the Bloch space on the polydisc, Abstr. Appl. Anal., vol. 2007, Article ID 48478, 2007, 12p.
[17]	Li, S.; Stević, S., Compactness of riemann – stieltjes operators between \(F(p, q, s)\) and \(\alpha\)-Bloch spaces, Publ. math. debrecen, 72, 1-2, 111-128, (2008) · Zbl 1164.47040
[18]	Li, S.; Stević, S., Generalized composition operators on Zygmund spaces and Bloch type spaces, J. math. anal. appl., 338, 1282-1295, (2008) · Zbl 1135.47021
[19]	Li, S.; Stević, S., Products of composition and integral type operators from \(H^\infty\) to the Bloch space, Complex variables elliptic funct., 53, 5, 463-474, (2008) · Zbl 1159.47019
[20]	Li, S.; Stević, S., Riemann – stieltjes operators between mixed norm spaces, Indian J. math., 50, 1, 177-188, (2008) · Zbl 1159.47012
[21]	Li, S.; Stević, S., Products of Volterra type operator and composition operator from \(H^\infty\) and Bloch spaces to the Zygmund space, J. math. anal. appl., 345, 40-52, (2008) · Zbl 1145.47022
[22]	S. Li, S. Stević, Weighted composition operators between \(H^\infty\) and \(\alpha\)-Bloch spaces in the unit ball, Taiwan. J. Math. 12 (2008).
[23]	Luo, L.; Ueki, S.I., Weighted composition operators between weighted Bergman spaces and Hardy spaces on the unit ball of \(\mathbb{C}^n\), J. math. anal. appl., 326, 1, 88-100, (2007) · Zbl 1114.32003
[24]	MacCluer, B.D.; Zhao, R., Essential norms of weighted composition operators between Bloch-type spaces, Rocky mountain J. math., 33, 4, 1437-1458, (2003) · Zbl 1061.30023
[25]	Madigan, K.; Matheson, A., Compact composition operators on the Bloch space, Trans. am. math. soc., 347, 7, 2679-2687, (1995) · Zbl 0826.47023
[26]	Ohno, S., Weighted composition operators between \(H^\infty\) and the Bloch space, Taiwanese J. math., 5, 555-563, (2001) · Zbl 0997.47025
[27]	Ohno, S.; Stroethoff, K.; Zhao, R., Weighted composition operators between Bloch-type spaces, Rocky mountain J. math., 33, 191-215, (2003) · Zbl 1042.47018
[28]	Rudin, W., Function theory in the unit ball of \(\mathbb{C}^n\), (1980), Springer-Verlag Berlin, Heidelberg, New York · Zbl 0495.32001
[29]	Shields, A.L.; Williams, D.L., Bounded projections, duality, and multipliers in spaces of analytic functions, Trans. am. math. soc., 162, 287-302, (1971) · Zbl 0227.46034
[30]	Stević, S., On an integral operator on the unit ball in \(\mathbb{C}^n\), J. inequal. appl., 1, 81-88, (2005) · Zbl 1074.47013
[31]	Stević, S., Boundedness and compactness of an integral operator on a weighted space on the polydisc, Indian J. pure appl. math., 37, 6, 343-355, (2006) · Zbl 1121.47032
[32]	Stević, S., Composition operators between \(H^\infty\) and the \(\alpha\)-Bloch spaces on the polydisc, Z. anal. anwendungen, 25, 4, 457-466, (2006) · Zbl 1118.47015
[33]	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
[34]	S. Stević, Weighted composition operators between mixed norm spaces and \(H_\alpha^\infty\) spaces in the unit ball, J. Inequal. Appl., Article ID 28629, 2007, 9pp. · Zbl 1138.47019
[35]	Stević, S., Norm of weighted composition operators from Bloch space to \(H_\mu^\infty\) on the unit ball, Ars. combin., 88, 125-127, (2008) · Zbl 1224.30195
[36]	S. Stević, On a new integral-type operator from the weighted Bergman space to the Bloch-type space on the unit ball, Discrete Dyn. Nat. Soc., Article ID 154263, 2008, 14pp.
[37]	S. Stević, The boundedness and compactness of an integral operator between \(H^\infty\) and a mixed norm space on the polydisc, Siberian J. Math., in press.
[38]	S. Stević, Products of integral type operators and composition operators from the mixed norm space to Bloch-type spaces, in press. · Zbl 1219.47050
[39]	Tang, X., Extended Cesàro operators between Bloch-type spaces in the unit ball of \(\mathbb{C}^n\), J. math. anal. appl., 326, 2, 1199-1211, (2007) · Zbl 1117.47022
[40]	S.I. Ueki, L. Luo, Compact weighted composition operators and multiplication operators between Hardy spaces, Abstr. Appl. Anal., Article ID 196498, 2008, 11pp. · Zbl 1167.47020
[41]	Yamashita, S., Gap series and \(\alpha\)-Bloch functions, Yokohama math. J., 28, 31-36, (1980) · Zbl 0467.30001
[42]	Ye, S., Weighted composition operators between the little \(\alpha\)-Bloch space and the logarithmic Bloch, J. comput. anal. appl., 10, 2, 243-252, (2008) · Zbl 1152.47019
[43]	Zhu, K., Spaces of holomorphic functions in the unit ball, Graduate texts in mathematics, (2005), Springer