Weighted composition operators on Hilbert function spaces on the ball

TL;DR

This paper investigates the properties of weighted composition operators on unitarily invariant Hilbert spaces on the ball, revealing a dichotomy in their unitary and co-isometric behavior across different spaces.

Contribution

It extends the classification of unitary weighted composition operators from the disc to higher-dimensional balls, identifying conditions for their existence.

Findings

Spaces with kernel (1 - ⟨z,w⟩)^(-γ) admit many unitary operators

Other spaces only admit trivial unitary operators

Results extend to infinite-dimensional cases

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 $\mathbb B_d$. We establish a dichotomy between the spaces $\mathcal{H}_\gamma$ with reproducing kernel $(1 - \langle z,w \rangle)^{-\gamma}$ 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 Mart\'in, Mas and Vukoti\'c from the disc to the ball. Some of our results continue to hold when $d = \infty$.