Unitary weighted composition operators on Bergman-Besov and Hardy Hilbert spaces on the ball
Hakkı Turgay KaptanoğluOn 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.
