Composition operators acting on weighted Dirichlet spaces
The paper is devoted to study composition operators acting on weighted Dirichlet spaces. Given an analytic function on the open unit disk in the complex plane \(\varphi: \mathbb{D}\rightarrow \mathbb{D}\), the associated composition operator \(C_{\varphi}\) acting on the space of holomorphic functions on the disk \(H(\mathbb{D})\) is defined by \(C_{\varphi}f(z)=f(\varphi(z))\). It is an interesting problem to describe the operator properties of \(C_{\varphi}\) in terms of the symbol \(\varphi\) when the operator acts on several spaces of analytic functions in \(\mathbb{D}\). They are well known for the Hardy and Bergman spaces. In the present paper, the authors focus on the weighted Dirichlet spaces \(\mathfrak{D}_{\alpha}\), \(0<\alpha<1\). They consist of those analytic functions \(f\) on \(\mathbb{D}\) with \[ \|f\|_{\mathfrak{D}_{\alpha}}={\left({|f(0)|}^2+\int_{\mathbb{D}}{|f'(z)|}^2\,dA_{\alpha}(z)\right)}^{1/2}<\infty, \] where \(dA_{\alpha}(z)=(1+\alpha)(1-|z|^2)^{\alpha}\,dA(z)\), and \(dA(z)=\frac{1}{\pi}\,dx\,dy\) is the normalized area measure on \(\mathbb{D}\). The continuity of the operator acting on these spaces was characterized in [\textit{K. Kellay} and \textit{P. Lefèvre}, J. Math. Anal. Appl. 386, No. 2, 718--727 (2012; Zbl 1231.47024)] and [\textit{N. Zorboska}, Proc. Am. Math. Soc. 126, No. 7, 2013--2023 (1998; Zbl 0894.47023)]. The authors estimate the essential norm of \(C_{\varphi}\) acting on \(\mathfrak{D}_{\alpha}\), i.e., the distance to the compact operators. They obtain a description of the membership in the Schatten-von~Neumann ideal \(S_p\) of \(\mathfrak{D}_{\alpha}\), that is, the compact operators \(C_{\varphi}\) such that its sequence of singular numbers is in the sequence space \(\ell_p\). Recall that the singular numbers of a compact operator \(T\) are the square root of the eigenvalues of the positive operator \(T^*T\), where \(T^*\) denotes the adjoint of \(T\). This description is done using a relationship of composition operators and Toeplitz operators. Analogously, they study the same properties of the composition operator acting from one weighted Dirichlet space \(\mathfrak{D}_{\alpha}\) to another Dirichlet space \(\mathfrak{D}_{\beta}\) with \(\alpha\neq \beta\). Finally, they characterize the composition operators with closed range. All this is obtained with the aid of the generalized Nevanlinna counting function of \(\varphi\).
scientific article; zbMATH DE number 6156276