Unitary weighted composition operators on Bergman-Besov and Hardy Hilbert spaces on the ball

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

References

1. Alpay, D., Kaptanoğlu, H.T., 2007, Toeplitz operators on Arveson and Dirichlet spaces, Integral Equations Operator Theory, 58, 1-33. google scholar
2. Beatrous, F., Burbea, J., 1989, Holomorphic Sobolev spaces on the ball, Dissertationes Math., 276, 57 pp. google scholar
3. Cowen, C.C., MacCluer, B.D., 1991, Composition Operators on Spaces of Analytic Functions, Stud. Adv. Math., CRC Press, Boca Raton. google scholar
4. Kaptanoğlu, H.T., 2005, Bergman projections on Besov spaces on balls, Illinois J. Math., 49, 385-403. google scholar
5. 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
6. 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
7. Rudin, W., 1980, Function Theory in the Unit Ball of C", Grundlehren Math. Wiss., vol. 241, Springer, New York. google scholar
8. Zhu, K., 2005, Spaces of Holomorphic Functions in the Unit Ball, Grad. Texts in Math., vol. 226, Springer, New York. google scholar