International Journal of Nonlinear Analysis and Applications

Norm of difference of two general polynomial weighted differentiation composition operators from Cauchy transform space into some analytic function ‎spaces

Articles in Press

Document Type : Research Paper

Authors

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

2 Engineering Faculty of Khoy‎, ‎Urmia University of Technology‎, ‎Urmia‎, ‎Iran

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

Abstract

 Let for $0\leq j\leq n$, $\varphi_j,\psi_j: \mathbb{D}\rightarrow\mathbb{D}$ and $u_j, v_j: \mathbb{D}\rightarrow\mathbb{C}$.  In this paper, we investigate boundedness of operator $$S =\sum_{j=0}^n  (D^j_{u_j, \varphi_j}-D^j_{v_j,\psi_j})$$ from Cauchy transform space into some analytic function spaces. Also, we obtain an exact formula for the norm of this operator.

Keywords

[1] E. Abbasi, S. Li, and H. Vaezi, Weighted composition operators from the Bloch space to nth weighted-type spaces, Turk. J. Math. 44 (2020), no. 1, 108–117.
[2] E. Abbasi, Y. Liu, and M. Hassanlou, Generalized Stevic-Sharma type operators from Hardy spaces into nth weighted type spaces, Turk. J. Math. 45 (2021), no. 4, 1543–1554.
[3] E. Abbasi and M. Hassanlou, Generalized Stevic-Sharma type operators on spaces of fractional Cauchy transforms, Mediterr. J. Math. 21 (2024), no. 40, 1–11.
[4] J. Cima and T.H. MacGregor, Cauchy Transforms of measures and univalent functions, Berenstein, C.A. (eds) Complex Analysis I. Lecture Notes in Mathematics, vol 1275, Springer, Berlin, Heidelberg, 1987.
[5] J. Cima and A. Matheson, Cauchy transforms and composition operators, Illinois J. Math. 42 (1998), no. 1, 58–69.
[6] J. Cima, A. Matheson, and W.T. Ross, The Cauchy Transform, Mathematical Surveys and Monographs. 125. Providence, RI: American Mathematical Society, 2006.
[7] X. Guo and M. Wang, Difference of weighted composition operators on the space of Cauchy integral transforms, Taiwanese J. Math. 22 (2018), 1435–1450.
[8] X. Guo and M. Wang, Linear combination of composition operators on Cauchy transform type spaces, Sci. China Math. 50 (2020), 1733–1744.
[9] R.A. Hibschweiler and T.H. MacGregor, Fractional Cauchy Transforms, Chapman and Hall. Boca Raton, FL: CRC, 2006.
[10] R.A. Hibschweiler, Composition operators on spaces of fractional Cauchy transforms, Complex Anal. Oper. Theory 6 (2012), 897–911.
[11] T.H. MacGregor, Fractional Cauchy transforms, J. Comput. Appl. Math. 105 (1999), no. 1-2, 93–108.
[12] A. Sharma and R. Krishan, Difference of composition operators from the space of Cauchy integral transforms to the Dirichlet space, Complex Anal. Oper. Theory 10 (2016), 141–152.
[13] A. Sharma, R. Krishan, and E. Subhadarsini, Difference of composition operators from the space of Cauchy integral transforms to Bloch-type spaces, Integral Transforms Spec. Funct. 28 (2017), 145–155.
[14] M. Wang and X. Guo, Difference of differentiation composition operators on the fractional Cauchy transforms spaces, Numer. Funct. Anal. Optim. 39 (2018), 1291–1315.
[15] K. Zhu, Operator Theory in Function Spaces, Pure and Applied Mathematics, Marcel Dekker, Inc., New York and Basel, 1990.
[16] X. Zhu, E. Abbasi, and D. Molaei, Weighted composition operators from the Besov space into nth weighted type spaces, Math. Slovaca 72 (2022), no. 4, 977–992.
[17] X. Zhu, Q. Hu, and D. Qu, Polynomial differentiation composition operators from Besov-type spaces into Bloch-type spaces, Math. Meth. Appl. Sci. 47 (2024), no. 1, 147–168.
International Journal of Nonlinear Analysis and Applications

Articles in Press, Corrected Proof
Available Online from 04 July 2025
  • Receive Date: 19 December 2024
  • Revise Date: 06 March 2025
  • Accept Date: 04 April 2025