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

Boundedness of weighted composition operators $W_{u,\phi }$ acting on the classical Dirichlet space $\mathcal{D}$ as $W_{u,\varphi}f= u, (f\circ \varphi)$isstudiedintermsofthemultiplierspaceassociatedtothesymbol$\varphi$,i\mathrm{.}e\mathrm{.},${$\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 $W_{u,\phi }$ in $\mathcal{D}$ whenever $\phi$ 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.