Bounded composition operators with closed range on the Dirichlet space

Abstract:

For composition operators on spaces of analytic functions it is well known that norm estimates can be converted to Carleson measure estimates. The boundedness of the composition operator becomes equivalent to a Carleson measure inequality. The measure corresponding to a composition operator $C_\varphi$ on the Dirichet space $\mathcal D$ is $d\nu _\varphi = n_\varphi dA$, where $n_\varphi (z)$ is the cardinality of the preimage $\varphi ^{-1}(z)$. The composition operator will have closed range if and only if the corresponding measure satisfies a “reverse Carleson measure” theorem: $\| f \|_{\mathcal {D}}^2 \le \int |f’|^2 d\nu _\varphi$ for all $f\in \mathcal D$. Assuming $C_\varphi$ is bounded, a necessary condition for this inequality is a reverse of the Carleson condition: (C) $\nu _\varphi (S) \ge c |S|$ for all Carleson squares $S$. It has long been known that this is not sufficient for a completely general measure. Here we show that it is also not sufficient for the special measures $\nu _\varphi$. That is, we construct a function $\varphi$ such that $C_\varphi$ is bounded and $\nu _\varphi$ satisfies (C) but the composition operator $C_\varphi$ does not have closed range.