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Composition operators on Dirichlet-type spaces

Authors:	R. A. Hibschweiler

Affiliation:	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 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

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