Essential norm estimates of generalized weighted composition operators into weighted type spaces

Subject Areas : Statistics

A. H. Sanatpour ^{1 *} , M. Hassanlou ²

1 - Department of Mathematics, Faculty of Mathematical Sciences and Computer, Kharazmi University
2 - Urmia University, Urmia, Iran.

Received: 2018-11-20 Accepted : 2019-12-21 Published : 2020-01-21

Keywords: فضاهای زیگموند, فضاهای از نوع وزن‌دار, عملگرهای ترکیبی وزن‌دار تعمیم‌یافته, نرم اساسی, فضاهای بلاخ,

Abstract :

Weighted composition operators appear in the study of dynamical systems and also in characterizing isometries of some classes of Banach spaces. One of the most important generalizations of weighted composition operators, are generalized weighted composition operators which in special cases of their inducing functions give different types of well-known operators like: weighted composition operators, composition operators, multiplication operators and composition operators followed by differentiation operators. In this paper we study generalized weighted composition operators and give estimates for the essential norm of such operators on certain Banach spaces of analytic functions into weighted type spaces. The underlying Banach spaces of analytic functions include Bloch spaces, Zygmund spaces and weighted type spaces. Our estimates for the essential norms of generalized weighted composition operators imply necessary and sufficient conditions for the compactness of such operators. As another application of our results, we obtain essential norm estimates of certain well-known operators which are special cases of generalized weighted composition operators.

References:

[1] Cowen, C. C., & MacCluer, B. D. (1995). Composition operators on spaces of analytic functions. CRC Press, Boca Raton.

 

[2] Shapiro, J. H. (1993). Composition operator and classical function theory. Springer-Verlag, New York.

 

[3] Yang, W., & Zhu, X. (2014). Differences of generalized weighted composition operators between growth spaces. Annales Polonici Mathematici.

 

[4] Li, S., & Stevic, S. (2015). Generalized weighted composition operators from -Bloch spaces into weighted-type spaces. Journal of Inequalities and Applications.

 

[5] Sanatpour, A. H., & Hassanlou, M. (2017). Essential norms of weighted differentiation composition operators between Zygmund type spaces and Bloch type spaces. Filomat.

 

[6] Stevic, S. (2008). Essential norms of weighted composition operators from the -Bloch space to a weighted-type space on the unit ball. Abstract and Applied Analysis.

 

[7] Zhu, X. (2015). Generalized weighted composition operators on Bloch-type spaces. Journal of Inequalities and Applications.

 

[8] Zhu, X. (2016). Essential norm of generalized weighted composition operators on Bloch-type spaces. Applied Mathematics and Computation.

 

[9] Zhu, X. (2012). Generalized weighted composition operators from Bloch spaces into Bers-type spaces. Filomat.

 

[10] Colonna, F., & Li, S. (2013). Weighted composition operators from the Bloch space and the analytic Besov spaces into the Zygmund space. Journal of Operators.

 

[11] Colonna, F., & Li, S. (2013). Weighted composition operators from  into the Zygmund spaces. Complex Analysis and Operator Theory.

 

[12] Liu, Y., & Yu, Y. (2012). Composition followed by differentiation between  and Zygmund spaces. Complex Analysis and Operator Theory.

 

[13] Esmaeili, K., & Lindstrom, M. (2013). Weighted composition operators between Zygmund type spaces and their essential norms. Integral Equations and Operator Theory.

 

[14] Hu, Q., & Ye, S. (2012). Weighted composition operators on the Zygmund spaces. Abstract and Applied Analysis.

 

[15] Hu, Q., & Zhu, X. (2016). Essential norm of weighted composition operators from  to the Zygmund space. Journal of Nonlinear Sciences and Applications.

 

[16] Li, S., & Stevic, S. (2008). Weighted composition operators from Zygmund spaces into Bloch spaces. Applied Mathematics and Computation.

 

[17] Hu, Q., Shi, Y., Shi, Y., & Zhu, X. (2016). Essential norm of generalized weighted composition operators from the Bloch space to the Zygmund space. Journal of Inequalities and Applications.

 

[18] Zhu, K. (1993). Bloch type spaces of analytic functions. Rocky Mountain Journal of Mathematics.

 

[19] Zhu, K. (2005). Spaces of holomorphic functions in the unit ball. Springer-Verlag, New York.

 

[20] Liu, Y., & Yu, Y. (2012). Composition followed by differentiation between  and Zygmund spaces. Complex Analysis and Operator Theory.

 

[21] Ohno, S. (2006). Products of composition and differentiation between Hardy spaces. Bulletein of Australian Mathematical Society.

 

[22] Montes-Rodriguez, A. (2000). Weighted composition operators on weighted Banach spaces of analytic functions. Journal of the London Mathematical Society.

 

[23] Hyvarinen, O., Kemppainen, M., Lindstrom, M., Rautio, A., & Saukko, E. (2012). The essential norm of weighted composition operators on weighted Banach spaces of analytic functions. Integral Equations and Operator Theory.