Abstract
On weighted Bergman and Hardy Hilbert spaces on the unit ball of the complex 𝑁-space, we consider weighted compositon operators 𝑇𝜓 in which the composition is by an automorphism 𝜓 of the unit ball and the weight is a power of the Jacobian of 𝜓 in such a way that the operator is unitary. Assuming that the homogeneous expansion of an 𝑓 in one of these spaces contains only terms with total degree even (odd, respectively) and the homogeneous expansion of 𝑇𝜓 𝑓 contains only terms with total degree odd (even, respectively), we prove that 𝑓 is the zero function. We also find related operators on the remaining Bergman-Besov Hilbert spaces including the Drury-Arveson space and the Dirichlet space for which the same result holds. Our results generalize the results obtained in Montes-Rodríguez (2023) on three function spaces on the unit disc to a wider family of function spaces on the unit ball.
Keywords
Weighted composition operator unitary operator Bergman-Besov space Hardy space Dirichlet space Drury-Arveson space
Thanks
The author thanks A. Ersin Üreyen of Eskişehir Technical University for useful discussions and an anonymous referee for careful attention to details.
References
- Alpay, D., Kaptanoğlu, H.T., 2007, Toeplitz operators on Arveson and Dirichlet spaces, Integral Equations Operator Theory, 58, 1-33. google scholar
- Beatrous, F., Burbea, J., 1989, Holomorphic Sobolev spaces on the ball, Dissertationes Math., 276, 57 pp. google scholar
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Abstract
References
- Alpay, D., Kaptanoğlu, H.T., 2007, Toeplitz operators on Arveson and Dirichlet spaces, Integral Equations Operator Theory, 58, 1-33. google scholar
- Beatrous, F., Burbea, J., 1989, Holomorphic Sobolev spaces on the ball, Dissertationes Math., 276, 57 pp. google scholar
- Cowen, C.C., MacCluer, B.D., 1991, Composition Operators on Spaces of Analytic Functions, Stud. Adv. Math., CRC Press, Boca Raton. google scholar
- Kaptanoğlu, H.T., 2005, Bergman projections on Besov spaces on balls, Illinois J. Math., 49, 385-403. google scholar
- Kaptanoğlu , H.T., Üreyen, A.E., 2018, Precise inclusion relations among Bergman-Besov and Bloch-Lipschitz spaces and H'x on the unit ball of C*, Math. Nachr., 291, 2236-2251. google scholar
- Montes-Rodrfguez, A., 2023, Zeros under unitary weighted composition operators in the Hardy and Bergman spaces, Mediterr. J. Math., 20, #82, 9 pp. google scholar
- Rudin, W., 1980, Function Theory in the Unit Ball of C", Grundlehren Math. Wiss., vol. 241, Springer, New York. google scholar
- Zhu, K., 2005, Spaces of Holomorphic Functions in the Unit Ball, Grad. Texts in Math., vol. 226, Springer, New York. google scholar
Details
| Primary Language | English |
|---|---|
| Subjects | Pure Mathematics (Other) |
| Journal Section | Research Articles |
| Authors | |
| Publication Date | June 21, 2023 |
| Published in Issue | Year 2023 |