## Essential norm of generalized weighted composition operators from the Bloch space to the Zygmund space.(English)Zbl 1337.30066

Summary: In this paper, we give some estimates of the essential norm for generalized weighted composition operators from the Bloch space to the Zygmund space. Moreover, we give a new characterization for the boundedness and compactness of the operator.

### MSC:

 30H30 Bloch spaces 47B33 Linear composition operators

### Keywords:

Bloch space; Zygmund space; composition operators
Full Text:

### References:

 [1] Hibschweiler, R; Portnoy, N, Composition followed by differentiation between Bergman and Hardy spaces, Rocky Mt. J. Math., 35, 843-855, (2005) · Zbl 1079.47031 [2] Li, S; Stević, S, Composition followed by differentiation between Bloch type spaces, J. Comput. Anal. Appl., 9, 195-205, (2007) · Zbl 1132.47026 [3] Li, S; Stević, S, Composition followed by differentiation between $$H^{∞}$$ and $$α$$-Bloch spaces, Houst. J. Math., 35, 327-340, (2009) · Zbl 1166.47034 [4] Li, S; Stević, S, Products of composition and differentiation operators from Zygmund spaces to Bloch spaces and bers spaces, Appl. Math. Comput., 217, 3144-3154, (2010) · Zbl 1204.30046 [5] Stević, S, Norm and essential norm of composition followed by differentiation from $$α$$-Bloch spaces to $$H^{∞}_{μ}$$, Appl. Math. Comput., 207, 225-229, (2009) · Zbl 1157.47026 [6] Stević, S, Products of composition and differentiation operators on the weighted Bergman space, Bull. Belg. Math. Soc. Simon Stevin, 16, 623-635, (2009) · Zbl 1181.30031 [7] Stević, S, Composition followed by differentiation from $$H^{∞}$$ and the Bloch space to $$n$$th weighted-type spaces on the unit disk, Appl. Math. Comput., 216, 3450-3458, (2010) · Zbl 1195.30070 [8] Stević, S, Characterizations of composition followed by differentiation between Bloch-type spaces, Appl. Math. Comput., 218, 4312-4316, (2011) · Zbl 1244.30082 [9] Yang, W, Products of composition differentiation operators from $$\mathcal{Q}_{K}(p,q)$$ spaces to Bloch-type spaces, Abstr. Appl. Anal., 2009, (2009) · Zbl 1185.47033 [10] Stević, S; Sharma, A, Iterated differentiation followed by composition from Bloch-type spaces to weighted BMOA spaces, Appl. Math. Comput., 218, 3574-3580, (2011) · Zbl 1253.30085 [11] Wu, Y; Wulan, H, Products of differentiation and composition operators on the Bloch space, Collect. Math., 63, 93-107, (2012) · Zbl 1267.30087 [12] Li, H; Fu, X, A new characterization of generalized weighted composition operators from the Bloch space into the Zygmund space, J. Funct. Spaces Appl., 2013, (2013) · Zbl 1383.47003 [13] Li, S; Stević, S, Generalized weighted composition operators from $$α$$-Bloch spaces into weighted-type spaces, J. Inequal. Appl., 2015, (2015) · Zbl 1338.47018 [14] Stević, S, Weighted differentiation composition operators from mixed-norm spaces to weighted-type spaces, Appl. Math. Comput., 211, 222-233, (2009) · Zbl 1165.30029 [15] Stević, S, Weighted differentiation composition operators from mixed-norm spaces to the $$n$$th weighted-type space on the unit disk, Abstr. Appl. Anal., 2010, (2010) · Zbl 1198.30014 [16] Stević, S, Weighted differentiation composition operators from $$H^{∞}$$ and Bloch spaces to $$n$$th weighted-type spaces on the unit disk, Appl. Math. Comput., 216, 3634-3641, (2010) · Zbl 1195.30073 [17] Yang, W; Zhu, X, Generalized weighted composition operators from area Nevanlinna spaces to Bloch-type spaces, Taiwan. J. Math., 16, 869-883, (2012) · Zbl 1268.47034 [18] Zhu, X, Products of differentiation, composition and multiplication from Bergman type spaces to bers type space, Integral Transforms Spec. Funct., 18, 223-231, (2007) · Zbl 1119.47035 [19] Zhu, X, Generalized weighted composition operators on weighted Bergman spaces, Numer. Funct. Anal. Optim., 30, 881-893, (2009) · Zbl 1183.47030 [20] Zhu, X, Generalized weighted composition operators on Bloch-type spaces, J. Inequal. Appl., 2015, (2015) · Zbl 1309.47027 [21] Zhu, X, Essential norm of generalized weighted composition operators on Bloch-type spaces, Appl. Math. Comput., 274, 133-142, (2016) · Zbl 1410.30032 [22] 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 [23] Li, S; Stević, S, Products of composition and integral type operators from $$H^{∞}$$ to the Bloch space, Complex Var. Elliptic Equ., 53, 463-474, (2008) · Zbl 1159.47019 [24] Li, S; Stević, S, Composition followed by differentiation from mixed-norm spaces to $$α$$-Bloch spaces, Sb. Math., 199, 1847-1857, (2008) · Zbl 1169.47025 [25] Li, S; Stević, S, Products of integral-type operators and composition operators between Bloch-type spaces, J. Math. Anal. Appl., 349, 596-610, (2009) · Zbl 1155.47036 [26] Stević, S, On an integral-type operator from logarithmic Bloch-type and mixed-norm spaces to Bloch-type spaces, Nonlinear Anal. TMA, 71, 6323-6342, (2009) · Zbl 1186.47033 [27] Stević, S, Products of integral-type operators and composition operators from the mixed norm space to Bloch-type spaces, Sib. Math. J., 50, 726-736, (2009) · Zbl 1219.47050 [28] Stević, S, On some integral-type operators between a general space and Bloch-type spaces, Appl. Math. Comput., 218, 2600-2618, (2011) · Zbl 1245.45011 [29] Stević, S; Sharma, A; Bhat, A, Products of multiplication composition and differentiation operators on weighted Bergman spaces, Appl. Math. Comput., 217, 8115-8125, (2011) · Zbl 1218.30152 [30] Stević, S; Sharma, A; Bhat, A, Essential norm of products of multiplication composition and differentiation operators on weighted Bergman spaces, Appl. Math. Comput., 218, 2386-2397, (2011) · Zbl 1244.30080 [31] Zhu, K: Operator Theory in Function Spaces, 2nd edn. Am. Math. Soc., Providence (2007) · Zbl 1123.47001 [32] Cowen, C, Maccluer, B: Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton (1995) · Zbl 0873.47017 [33] Hyvärinen, O; Lindström, M, Estimates of essential norm of weighted composition operators between Bloch-type spaces, J. Math. Anal. Appl., 393, 38-44, (2012) · Zbl 1267.47040 [34] Li, S; Stević, S, Weighted composition operators from Bergman-type spaces into Bloch spaces, Proc. Indian Acad. Sci. Math. Sci., 117, 371-385, (2007) · Zbl 1130.47016 [35] Li, S; Stević, S, Weighted composition operators between $$H^{∞}$$ and $$α$$-Bloch spaces in the unit ball, Taiwan. J. Math., 12, 1625-1639, (2008) · Zbl 1177.47032 [36] Li, S; Stević, S, Weighted composition operators from Zygmund spaces into Bloch spaces, Appl. Math. Comput., 206, 825-831, (2008) · Zbl 1215.47022 [37] MacCluer, B; Zhao, R, Essential norm of weighted composition operators between Bloch-type spaces, Rocky Mt. J. Math., 33, 1437-1458, (2003) · Zbl 1061.30023 [38] Madigan, K; Matheson, A, Compact composition operators on the Bloch space, Trans. Am. Math. Soc., 347, 2679-2687, (1995) · Zbl 0826.47023 [39] Manhas, J; Zhao, R, New estimates of essential norms of weighted composition operators between Bloch type spaces, J. Math. Anal. Appl., 389, 32-47, (2012) · Zbl 1267.47042 [40] Montes-Rodriguez, A, The essential norm of composition operators on the Bloch space, Pac. J. Math., 188, 339-351, (1999) · Zbl 0932.30034 [41] Ohno, S; Stroethoff, K; Zhao, R, Weighted composition operators between Bloch-type spaces, Rocky Mt. J. Math., 33, 191-215, (2003) · Zbl 1042.47018 [42] Stević, S, Essential norms of weighted composition operators from the $$α$$-Bloch space to a weighted-type space on the unit ball, Abstr. Appl. Anal., 2008, (2008) · Zbl 1160.32011 [43] Tjani, M: Compact composition operators on some Möbius invariant Banach space. PhD dissertation, Michigan State University (1996) · Zbl 1225.47036 [44] Wulan, H; Zheng, D; Zhu, K, Compact composition operators on BMOA and the Bloch space, Proc. Am. Math. Soc., 137, 3861-3868, (2009) · Zbl 1194.47038 [45] Zhao, R, Essential norms of composition operators between Bloch type spaces, Proc. Am. Math. Soc., 138, 2537-2546, (2010) · Zbl 1190.47028 [46] Choe, B; Koo, H; Smith, W, Composition operators on small spaces, Integral Equ. Oper. Theory, 56, 357-380, (2006) · Zbl 1114.47028 [47] Duren, P: Theory of $$H$$\^{}{$$p$$} Spaces. Academic Press, New York (1970) · Zbl 0215.20203 [48] Esmaeili, K; Lindström, M, Weighted composition operators between Zygmund type spaces and their essential norms, Integral Equ. Oper. Theory, 75, 473-490, (2013) · Zbl 1306.47036 [49] Li, S; Stević, S, Volterra type operators on Zygmund spaces, J. Inequal. Appl., 2007, (2007) · Zbl 1146.30303 [50] Yu, Y; Liu, Y, Weighted differentiation composition operators from $$H^{∞}$$ to Zygmund spaces, Integral Transforms Spec. Funct., 22, 507-520, (2011) · Zbl 1225.47036 [51] Montes-Rodriguez, A, Weighed composition operators on weighted Banach spaces of analytic functions, J. Lond. Math. Soc., 61, 872-884, (2000) · Zbl 0959.47016 [52] Hyvärinen, O; Kemppainen, M; Lindström, M; Rautio, A; Saukko, E, The essential norm of weighted composition operators on weighted Banach spaces of analytic functions, Integral Equ. Oper. Theory, 72, 151-157, (2012) · Zbl 1252.47026
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