Composition operators on Dirichlet-type spaces期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

按检索

Composition operators on Dirichlet-type spaces

Authors:	R A Hibschweiler

Institution:	Department of Mathematics, University of New Hampshire, Durham, New Hampshire 03824

Abstract:	The Dirichlet-type space $D^{p} (1 \leq p \leq 2$ ) is the Banach space of functions analytic in the unit disc with derivatives belonging to the Bergman space $A^{p}$ . Let $\Phi$ be an analytic self-map of the disc and define $C_{\Phi}(f) = f \circ \Phi$ for $f \in D^{p}$ . The operator $C_{\Phi}: D^{p} \rightarrow D^{p}$ is bounded (respectively, compact) if and only if a related measure $\mu_{p}$ is Carleson (respectively, compact Carleson). If $C_{\Phi}$ is bounded (or compact) on $D^{p}$ , then the same behavior holds on $D^{q} (1 \leq q < p$ ) and on the weighted Dirichlet space $D_{2-p}$ . Compactness on $D^{p}$ implies that $C_{\Phi}$ is compact on the Hardy spaces and the angular derivative exists nowhere on the unit circle. Conditions are given which, together with the angular derivative condition, imply compactness on the space $D^{p}$ . Inner functions which induce bounded composition operators on $D^{p}$ are discussed briefly.

Keywords:	Composition operator Dirichlet space Carleson measure angular derivative

	点击此处可从《Proceedings of the American Mathematical Society》浏览原始摘要信息
	点击此处可从《Proceedings of the American Mathematical Society》下载免费的PDF全文