On compactness of the difference of composition operators |
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Authors: | Pekka J Nieminen Eero Saksman |
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Affiliation: | a Department of Mathematics and Statistics, University of Helsinki, PO Box 68, FIN-00014 Helsinki, Finland b Department of Mathematics and Statistics, University of Jyväskylä, PO Box 35, FIN-40014 Jyväskylä, Finland |
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Abstract: | Let φ and ψ be analytic self-maps of the unit disc, and denote by Cφ and Cψ the induced composition operators. The compactness and weak compactness of the difference T=Cφ−Cψ are studied on Hp spaces of the unit disc and Lp spaces of the unit circle. It is shown that the compactness of T on Hp is independent of p∈1,∞). The compactness of T on L1 and M (the space of complex measures) is characterized, and examples of φ and ψ are constructed such that T is compact on H1 but non-compact on L1. Other given results deal with L∞, weakly compact counterparts of the previous results, and a conjecture of J.E. Shapiro. |
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Keywords: | Composition operator Aleksandrov measure Compactness Difference |
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