Mean ergodicity of multiplication operators in weighted Dirichlet spaces
Antonio Bonilla, Daniel Seco
TL;DR
The authors investigate bounded multiplication operators on weighted Dirichlet spaces $D_{\alpha}$ for $-1<\alpha<1$, characterizing when $M_{\phi}$ and its adjoint are power bounded, Cesàro bounded, uniformly Kreiss bounded, or mean ergodic in terms of the symbol $\phi$ and Carleson-measure conditions. A central result is that, in this range of $\alpha$, mean ergodicity and Cesàro boundedness are equivalent for these operators, with precise integral testing conditions and several sufficient criteria expressed through Carleson measures. They establish comprehensive criteria for PB, CB, ME, and UKB, and derive a notable example of a uniform mean ergodic multiplication operator on the Dirichlet space that is not power bounded, also correcting a mischaracterization in Besov spaces for the $p=2$ (Dirichlet) case. The paper also discusses the adjoints, presents both necessary and sufficient conditions, and ends with open problems and directions for extending the analysis to broader operator classes and spaces.
Abstract
We characterize bounded multiplication operators in weighted Dirichlet spaces that are power bounded, Cesàro bounded and uniformly Kreiss. Moreover, we show the equivalence in such spaces between mean ergodicity and Cesàro boundedness for multiplication operators. We perform the same study for adjoints of multiplication operators. As a particular example, we obtain a uniform mean ergodic multiplication operator in Dirichlet spaces that fails to be power bounded.