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

TL;DR

This paper investigates the boundedness and spectral properties of weighted composition operators on the Dirichlet space, emphasizing the role of multipliers and extending existing spectral results to this function space.

Contribution

It characterizes bounded weighted composition operators on the Dirichlet space using multipliers and extends spectral analysis results for automorphisms.

Findings

Boundedness characterized via multiplier spaces.

Spectral properties extended to automorphisms of the unit disc.

Connections established between multipliers and operator boundedness.

Abstract

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