## Essential norm of generalized weighted composition operators from $$H^\infty$$ to the logarithmic Bloch space.(English)Zbl 1421.30075

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

### MSC:

 30H30 Bloch spaces 47B33 Linear composition operators
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### References:

 [1] F. Colonna and S. Li, Weighted composition operators from Hardy spaces into logarithmic Bloch spaces, J. Funct. Spaces Appl. 2012 (2012), article ID 454820. · Zbl 1250.47027 [2] C. Cowen and B. Maccluer, Composition operators on spaces of analytic functions, CRC Press, Boca Raton, FL, 1995. · Zbl 0873.47017 [3] R. Hibschweiler and N. Portnoy, Composition followed by differentiation between Bergman and Hardy spaces, Rocky Mountain J. Math. 35 (2005), 843-855. · Zbl 1079.47031 [4] O. Hyvärinen, M. Kemppainen, M. Lindström, A. Rautio and E. Saukko, The essential norm of weighted composition operators on weighted Banach spaces of analytic functions, Int. Eqs. Oper. Th. 72 (2012), 151-157. · Zbl 1252.47026 [5] O. Hyvärinen and M. Lindström, Estimates of essential norm of weighted composition operators between Bloch-type spaces, J. Math. Anal. Appl. 393 (2012), 38-44. [6] H. Li and X. Fu, A new characterization of generalized weighted composition operators from the Bloch space into the Zygmund space, J. Funct. Spaces Appl. 2013 (2013), article ID 925901. · Zbl 1383.47003 [7] S. Li and S. Stević, Composition followed by differentiation between Bloch type spaces, J. Comp. Anal. Appl. 9 (2007), 195-205. · Zbl 1132.47026 [8] —-, Composition followed by differentiation from mixed-norm spaces to $$\a$$-Bloch spaces, Sbor. Math. 199 (2008), 1847-1857. · Zbl 1169.47025 [9] —-, Composition followed by differentiation between $$H^\infty$$ and $$\alpha$$-Bloch spaces, Houston J. Math. 35 (2009), 327-340. · Zbl 1166.47034 [10] —-, Products of composition and differentiation operators from Zygmund spaces to Bloch spaces and Bers spaces, Appl. Math. Comp. 217 (2010), 3144-3154. · Zbl 1204.30046 [11] B. Maccluer and R. Zhao, Essential norm of weighted composition operators between Bloch-type spaces, Rocky Mountain J. Math. 33 (2003), 1437-1458. · Zbl 1061.30023 [12] K. Madigan and A. Matheson, Compact composition operators on the Bloch space, Trans. Amer. Math. Soc. 347 (1995), 2679-2687. · Zbl 0826.47023 [13] J. Manhas and R. Zhao, New estimates of essential norms of weighted composition operators between Bloch type spaces, J. Math. Anal. Appl. 389 (2012), 32-47. · Zbl 1267.47042 [14] A. Montes-Rodriguez, Weighed composition operators on weighted Banach spaces of analytic functions, J. Lond. Math. Soc. 61 (2000), 872-884. · Zbl 0959.47016 [15] S. Ohno, K. Stroethoff and R. Zhao, Weighted composition operators between Bloch-type spaces, Rocky Mountain J. Math. 33 (2003), 191-215. · Zbl 1042.47018 [16] S. Stević, Generalized composition operators from logarithmic Bloch spaces to mixed-norm spaces, Utilitas Math. 77 (2008), 167-172. [17] —-, On a new operator from the logarithmic Bloch space to the Bloch-type space on the unit ball, Appl. Math. Comp. 206 (2008), 313-320. · Zbl 1162.47029 [18] —-, Norm and essential norm of composition followed by differentiation from $$\alpha$$-Bloch spaces to $$H^\infty_\mu$$, Appl. Math. Comp. 207 (2009), 225-229. · Zbl 1157.47026 [19] —-, On an integral-type operator from logarithmic Bloch-type and mixed-norm spaces to Bloch-type spaces, Nonlin. Anal. 71 (2009), 6323-6342. · Zbl 1186.47033 [20] —-, Products of composition and differentiation operators on the weighted Bergman space, Bull. Belgian Math. Soc. 16 (2009), 623-635. · Zbl 1181.30031 [21] S. Stević, Weighted differentiation composition operators from mixed-norm spaces to weighted-type spaces, Appl. Math. Comp. 211 (2009), 222-233. [22] —-, On operator $$P_\varphi^g$$ from the logarithmic Bloch-type space to the mixed-norm space on unit ball, Appl. Math. Comp. 215 (2010), 4248-4255. · Zbl 1205.45014 [23] —-, Weighted differentiation composition operators from mixed-norm spaces to the $$n$$-th weighted-type space on the unit disk, Abstracts Appl. Anal. 2010 (2010), article ID 246287. · Zbl 1198.30014 [24] —-, Weighted differentiation composition operators from $$H^\infty$$ and Bloch spaces to $$n$$-th weigthed-type spaces on the unit disk, Appl. Math. Comp. 216 (2010), 3634-3641. · Zbl 1195.30073 [25] M. Tjani, Compact composition operators on some Möbius invariant Banach spaces, Ph.D. dissertation, Michigan State University, East Lansing, 1996. [26] Y. Wu and H. Wulan, Products of differentiation and composition operators on the Bloch space, Collect. Math. 63 (2012), 93-107. · Zbl 1267.30087 [27] H. Wulan, D. Zheng and K. Zhu, Compact composition operators on BMOA and the Bloch space, Proc. Amer. Math. Soc. 137 (2009), 3861-3868. · Zbl 1194.47038 [28] W. Yang and X. Zhu, Generalized weighted composition operators from area Nevanlinna spaces to Bloch-type spaces, Taiwanese J. Math. 16 (2012), 869-883. · Zbl 1268.47034 [29] R. Zhao, Essential norms of composition operators between Bloch type spaces, Proc. Amer. Math. Soc. 138 (2010), 2537-2546. · Zbl 1190.47028 [30] K. Zhu, Operator theory in function spaces, Marcel Dekker, New York, 1990. [31] X. Zhu, Products of differentiation, composition and multiplication from Bergman type spaces to Bers type space, Int. Tran. Special Funct. 18 (2007), 223-231. · Zbl 1119.47035 [32] —-, Generalized weighted composition operators on weighted Bergman spaces, Num. Funct. Anal. Optim. 30 (2009), 881-893. · Zbl 1183.47030 [33] —-, Generalized weighted composition operators on Bloch-type spaces, J. Inequal. Appl. 2015 (2015), 59-68. · Zbl 1309.47027 [34] —-, Generalized weighted composition operators from $$H^\infty$$ to the logarithmic Bloch space, Filomat 30 (2016), 3867-3874. · Zbl 1465.47019 [35] —-, Essential norm of generalized weighted composition operators on Bloch-type spaces, Appl. Math. Comp. 274 (2016), 133-142. · Zbl 1410.30032
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