Composition operators related to the Dirichlet space (Q475797)

For an analytic self-map \(\varphi\) of the unit disk \(\mathbb{D}\), the composition operator \(C_{\varphi}\) induced by \(\varphi\) is defined by \(C_{\varphi}f=f\circ\varphi\), where \(f\) is analytic in \(\mathbb{D}\). Properties of composition operators \(C_{\varphi}: H^2 \rightarrow H^2\), \(C_{\varphi}: {\mathcal{D}} \rightarrow {\mathcal{D}}\), and \(C_{\varphi}: {\mathcal{D}} \rightarrow H^2\) have been intensively investigated in the past two decades. It is well known that the composition operator \(C_{\varphi}\) is always bounded on the Hardy spaces \(H^p\) \((0<p<\infty)\), and hence on the Dirichlet space \(\mathcal{D}\). Hilbert-Schmidtness of \(C_{\varphi}\) acting on each and between the spaces \(\mathcal{D}\) and \(H^2\) have been characterized by \textit{J. H. Shapiro} and \textit{P. D. Taylor} in [Indiana Univ. Math. J. 23, 471--496 (1973; Zbl 0276.47037)]. In this paper, the author investigates the open questions of boundedness and compactness of composition operators acting between \(H^2\) and \(\mathcal{D}\). In the first part (Section 2), the author uses the standard weak-convergence criterion to prove that \(C_{\varphi}: {\mathcal{D}} \rightarrow H^2\) is always compact. It is pointed out that composition operators induced by inner functions serve as examples of bounded but not Hilbert-Schmidt, in sharp contrast to the case of \(H^2\), when \(C_{\varphi}\), acting from \(\mathcal{D}\) to \(H^2\), satisfies the Hilbert-Schmit criterion, where \(\varphi(z)=(z+2)/2\). For \(h > 0\), \(\lambda \in \mathbb{D}\), let \(S(\theta,h)\) represent a Carleson square at \(e^{i\theta} \in \partial \mathbb{D}\), \(\alpha_{\lambda}(z)=(\lambda-z)/(1-\overline{\lambda}z)\), and \(d\mu=\eta_{\varphi}(\omega)dA\), where \(\eta_{\varphi}(\omega)\) represents the cardinality of \(\varphi^{-1}(\omega)\). In the second part (Section 3), the author proves characterizations of boundedness and compactness of \(C_{\varphi}\) acting from \(H^2\) to \(\mathcal{D}\). More specifically, the author shows the equivalence of the following conditions:{\parindent=0,5cm \begin{itemize}\item[(i)] \(C_{\varphi}\) maps \(H^2\) boundedly (resp. compactly) into \(\mathcal{D}\),\item[(ii)] \(\mu(S(\theta,h)) \leq Ch^3\) for some constant \(C > 0\) (resp., \(\lim\limits_{h \rightarrow 0}\sup\limits_{\theta \in [0,\pi]}\frac{ \mu(S(\theta,h))}{h^3} = 0)\), \item[(iii)] there exists some constant \(C > 0\) such that \(\int_{\mathbb{D}}|\alpha'_{\lambda}(z)|^3d\mu(z) \leq C\) for all \(\lambda \in \mathbb{D}\) (resp., \(\lim\limits_{|\lambda| \rightarrow 1}\int_{\mathbb{D}}|\alpha'_{\lambda}(z)|^3d\mu(z) = 0)\), \item [(iv)] \(\sup\limits_{\lambda \in \mathbb{D}}\int_{\mathbb{D}}\frac{|\varphi'(z)|^2}{(1-|\varphi(z)|^2)^3}(1-|\alpha_{\lambda}(\varphi(z))|^2)^3dA(z) < \infty\) (resp., \(\lim\limits_{|\lambda| \rightarrow 1}\int_{\mathbb{D}}\frac{|\varphi'(z)|^2}{(1-|\varphi(z)|^2)^3}(1-|\alpha_{\lambda}(\varphi(z))|^2)^3dA(z)=0)\). \end{itemize}} As an illustration of these results, the author uses condition (ii) of the boundedness result to construct an example of \(C_{\varphi}\) from \(H^2\) to \(\mathcal{D}\) which is bounded, and shows that \(C_{\varphi}\) is not bounded for \(\phi(z)=(z+1)/2\), proving that not all composition operators boundedly map \(H^2\) to \(\mathcal{D}\). The author also uses the condition (ii) for compactness, to construct an example of \(C_{\varphi}\) which maps \(H^2\) compactly into \(\mathcal{D}\). The paper sets the stage for the investigations of extending the results to composition and weighted composition operators acting between classical Hilbert, Bergman, and Dirichlet spaces.