A class of operator related weighted composition operators between Zygmund space

Document Type : Original Article

Author

Department of Mathematics‎, ‎Mahabad Branch‎, ‎Islamic Azad University‎, ‎Mahabad‎, ‎Iran

10.22060/ajmc.2020.18833.1041

Abstract

Let D be the open unit disk in the complex plane C and H(D) be the set of all analytic functions on D. Let u, v ∈ H(D) and ϕ be an analytic self-map of D. A class of operator related weighted composition operators is defined as follow
Tu,v,ϕf(z) = u(z)f(ϕ(z)) + v(z)f 0 (ϕ(z)), f ∈ H(D), z ∈ D.
In this work, we obtain some new characterizations for boundedness and essential norm of operator Tu,v,ϕ between Zygmund space.

Keywords

Main Subjects

References

[1] C. C. Cowen, B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, FL, 1995.
[2] K. Esmaeili, M. Lindstr¨om, Weighted composition operators between Zygmund type spaces and their essential norms, Integr. Equ. Oper. Theory, 75 (2013) 473-490.
[3] Z. Jiang, Product-type operators from Zygmund spaces to Bloch-Orlicz spaces, Complex Var. Elliptic Equ., 62 (11) (2017) 1645-1664.
[4] Y. Liu, Y. Yu, On a Stevi´c-Sharma operator from Hardy spaces to the logarithmic Bloch spaces, J. Inequal. Appl., 2015 (2015) 2015:22, DOI 10.1186/s13660-015-0547-1.
[5] Y. Liu, Y. Yu, Products of composition, multiplication and radial derivative operators from logarithmic Bloch spaces to weighted-type spaces on the unit ball, J. Math. Anal. Appl., 423 (1) (2015) 76-93.
[6] Y. Liu, Y. Yu, On an extension of Stevi´c-Sharma operator from the general space to weighted-type spaces on the unit ball, Complex Anal. Oper. Theory, 11 (2) (2017) 261-288.
[7] Y. Liu, Y. Yu, On Stevi´c-Sharma type operators from the Besov spaces into the weighted-type space H∞ µ , Math. Inequal. Appl., 22 (3) (2019) 1037-1053.
[8] B. D. Maccluer, R. Zhao, Essential norms of weighted composition operators between Bloch-type spaces, Rocky Mountain J. Math., 33 (2003) 1437-1458.
[9] S. Stevi´c, A. Sharma, A. Bhat, Products of multiplication composition and differentiation operators on weighted Bergman spaces, Appl. Math. Comput., 217 (2011) 81158125.
[10] S. Stevi´c, A. Sharma, A. Bhat, Essential norm of products of multiplications composition and differentiation operators on weighted Bergman spaces, Appl. Math. Comput., 218 (2011) 2386-2397.
[11] S. Stevi´c, A. Sharma, On a product-type operator between Hardy and α-Bloch spaces of the upper half-plane, J. Inequal. Appl., 2018:273, 2018.
[12] M. Tjani, Compact composition operators on some M¨obius invariant Banach space, PhD dissertation, Michigan State university, 1996.
[13] H. Wulan, D. Zheng, K. Zhu, Compact composition operators on BMOA and the Bloch space, Proc. Amer. Math. Soc., 137 (2009) 3861-3868.
[14] S. Y, Q. Hu, Weighted Composition Operators on the Zygmund Space, Abstract and Applied Analysis, 2012 (2012), Article ID 462482, 18 pages doi:10.1155/2012/462482.
[15] Y. Yu, Y. Liu, On Stevi´c type operator from H∞ space to the logarithmic Bloch spaces, Complex Anal. Oper. Theory, 9 (8) (2015) 1759-1780.
[16] F. Zhang, Y. Liu, On the compactness of the Stevi´c-Sharma operator on the logarithmic Bloch spaces, Math. Inequal. Appl., 19(2) (2016) 625-642.
[17] F. Zhang, Y. Liu, On a Stevi´c-Sharma operator from Hardy spaces to Zygmund-type spaces on the unit disk, Complex Anal. Oper. Theory, 12 (1) (2018) 81-100.
[18] K. Zhu, Bloch type spaces of analytic functions, Rocky Mountain J. Math., 23 (3) (1993) 1143-1177.
AUT Journal of Mathematics and Computing
Volume 2, Issue 1
Winter and Spring 2021
Pages 17-25

Files

History

  • Receive Date: 06 August 2020
  • Revise Date: 25 September 2020
  • Accept Date: 25 September 2020

Share

How to cite

Statistics