Computational Methods and Function Theory 7 (2007), No. 2, 325--344
Copyright Heldermann Verlag 2007
Pekka J. Nieminen
pjniemin@cc.helsinki.fi , University of Helsinki, Department of Mathematics and Statistics, P.O. Box 68, FI-00014 University of Helsinki, Finland.
We consider the difference T = C&phi-C&psi of two analytic composition operators in the unit disc. We characterize the compactness and weak compactness of T on the standard Bloch space, improving an earlier result by Hosokawa and Ohno. We also characterize the compactness and weak compactness of T on analytic Lipschitz spaces. These characterizations are derived from a general result dealing with differences of weighted composition operators on weighted Banach spaces of analytic functions. We also make complementary remarks on the compactness properties of a single composition operator on the Lipschitz spaces and answer a question of Cowen and MacCluer on the boundedness of such an operator.
Keywords: composition operator, compactness, difference, Bloch space, Lipschitz space
MSC 2000: Primary 47B33; Secondary 30D45, 47B07.
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