Isometric composition operators on weighted Dirichlet space

summary:We investigate isometric composition operators on the weighted Dirichlet space $\mathcal {D}_\alpha $ with standard weights $(1-|z|^2)^\alpha $, $\alpha >-1$. The main technique used comes from Martín and Vukotić who completely characterized the isometric composition operators on the classical Dirichlet space $\mathcal {D}$. We solve some of these but not in general. We also investigate the situation when $\mathcal {D}_\alpha $ is equipped with another equivalent norm