Abstract
References
- [1] A. Babaei and A. Abkar, Weighted composition-differentiation operators on the Hardy and Bergman spaces, Bull. Iran. Math. Soc. 48, 3637–3658, 2022.
- [2] G. Cao, L. He and K. Zhu, Polynomial approximation and composition operators, Proc. Amer. Math. Soc. 149 (9), 3715–3724, 2021.
- [3] J.A. Cima, A theorem on composition operators, Banach spaces of analytic functions (Proc. Pelczynski Conf., Kent State Univ., Kent, Ohio, 1976). Lecture Notes in Math. 604, Springer, Berlin, pp. 21–24, 1977.
- [4] C.C. Cowen and B.D. MacCluer, Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, 1995.
- [5] P. Duren, Theory of $H^p$ Spaces, Academic Press, New York, 1970.
- [6] M. Fatehi and C.N.B. Hammond, Composition-differentiation operators on the Hardy space, Proc. Amer. Math. Soc. 148 (7), 2893–2900, 2020.
- [7] M. Fatehi and C.N.B. Hammond, Normality and self-adjointness of weighted composition-differentiation operators, Complex Anal. Oper. Theory 15 9, 2021.
- [8] E. Nordgren, Composition operators, Can. J. Math. 20, 442–449, 1968.
- [9] S. Ohno, Products of composition and differentiation between Hardy spaces, Bull. Austr. Math. Soc. 73 (2), 235–243, 2006.
- [10] W. Rudin, Real and Complex Analysis, 3rd edition, McGraw-Hill, 1986.
- [11] Z. Saeidikia and A. Abkar, Composition operators on weighted Bergman spaces of polydisk, Bull. Iran. Math. Soc. 49 (31), 2023.
- [12] K. Zhu, Operator Theory in Function Spaces, Marcel Dekker Inc., New York, 1990.
Abstract
We study composition-differentiation operators acting on the Bergman and Dirichlet space of the open unit disk. We first characterize the compactness of composition-differentiation operator on weighted Bergman spaces. We shall then prove that for an analytic self-map $\varphi$ on the open unit disk $\mathbb{D}$, the induced composition-differentiation operator is bounded with dense range if and only if $\varphi$ is univalent and the polynomials are dense in the Bergman space on $\Omega:=\varphi(\mathbb{D})$.
Keywords
composition-differentiation operator Bergman space Dirichlet space compact operator dense-range operator
References
- [1] A. Babaei and A. Abkar, Weighted composition-differentiation operators on the Hardy and Bergman spaces, Bull. Iran. Math. Soc. 48, 3637–3658, 2022.
- [2] G. Cao, L. He and K. Zhu, Polynomial approximation and composition operators, Proc. Amer. Math. Soc. 149 (9), 3715–3724, 2021.
- [3] J.A. Cima, A theorem on composition operators, Banach spaces of analytic functions (Proc. Pelczynski Conf., Kent State Univ., Kent, Ohio, 1976). Lecture Notes in Math. 604, Springer, Berlin, pp. 21–24, 1977.
- [4] C.C. Cowen and B.D. MacCluer, Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, 1995.
- [5] P. Duren, Theory of $H^p$ Spaces, Academic Press, New York, 1970.
- [6] M. Fatehi and C.N.B. Hammond, Composition-differentiation operators on the Hardy space, Proc. Amer. Math. Soc. 148 (7), 2893–2900, 2020.
- [7] M. Fatehi and C.N.B. Hammond, Normality and self-adjointness of weighted composition-differentiation operators, Complex Anal. Oper. Theory 15 9, 2021.
- [8] E. Nordgren, Composition operators, Can. J. Math. 20, 442–449, 1968.
- [9] S. Ohno, Products of composition and differentiation between Hardy spaces, Bull. Austr. Math. Soc. 73 (2), 235–243, 2006.
- [10] W. Rudin, Real and Complex Analysis, 3rd edition, McGraw-Hill, 1986.
- [11] Z. Saeidikia and A. Abkar, Composition operators on weighted Bergman spaces of polydisk, Bull. Iran. Math. Soc. 49 (31), 2023.
- [12] K. Zhu, Operator Theory in Function Spaces, Marcel Dekker Inc., New York, 1990.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Research Article |
| Authors | |
| Early Pub Date | August 15, 2023 |
| Publication Date | June 27, 2024 |
| Published in Issue | Year 2024 Volume: 53 Issue: 3 |