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

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

Zbl 1155.32002
Full Text:

### 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
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.