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

Boundedness of weighted composition operators $W_{u,φ}$ acting on the classical Dirichlet space $\mathcal{D}$ as $W_{u,φ}f= u\, (f\circ φ)$ is studied in terms of the multiplier space associated to the symbol $φ$, i.e., ${\mathcal{M}(ϕ)}=\{ u \in {\mathcal D}: W_{u,ϕ} \hbox{ is bounded on } {\mathcal D} \}$. A prominent role is played by the multipliers of the Dirichlet space. As a consequence, the spectrum of $W_{u,φ}$ in $\mathcal{D}$ whenever $φ$ 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.

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