Weighted Composition Operators on Hilbert Function Spaces on the Ball

Abstract

A weighted composition operator on a reproducing kernel Hilbert space is given by a composition, followed by a multiplication. We study unitary and co-isometric weighted composition operators on unitarily invariant spaces on the Euclidean unit ball Bd. We establish a dichotomy between the spaces Hγ with reproducing kernel (1 − z, w ) −γ for γ > 0, and all other spaces. Whereas the former admit many unitary weighted composition operators, the latter only admit trivial ones. This extends results of Mart´ın, Mas and Vukoti´c from the disc to the ball. Some of our results continue to hold when d = ∞.