Weighted composition operators on the Dirichlet space: boundedness and spectral properties

Abstract

Boundedness of weighted composition operators $W_{u,\phi }$ acting on the classical Dirichlet space $\mathcal{D}$ as $W_{u,\phi }f=u(f\circ \phi )$ is studied in terms of the multiplier space associated to the symbol $\phi$, i.e., $\mathcal{M}(\varphi )=\{u\in \mathcal{D}:W_{u,\varphi }is\; bounded\; on\mathcal{D}\}$. A prominent role is played by the multipliers of the Dirichlet space. As a consequence, the spectrum of $W_{u,\phi }$ in $\mathcal{D}$ whenever $\phi$ is an automorphism of the unit disc is studied, extending a recent work of Hyvärinen, Lindström, Nieminen and Saukko to the context of the Dirichlet space.

15 pages. Accepted for publication in Math. Annalen