Weighted composition operators on Hilbert function spaces on the ball

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 ${\mathbb{B}}_d$. We establish a dichotomy between the spaces ${\mathcal{H}}_{\gamma }$ with reproducing kernel $(1-⟨z,w⟩{)}^{-\gamma }$ for $\gamma >0$, and all other spaces. Whereas the former admit many unitary weighted composition operators, the latter only admit trivial ones. This extends results of Martín, Mas and Vukotić from the disc to the ball. Some of our results continue to hold when $d=\mathrm{\infty }$.