Composition operators on small spaces

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    64 Citations (Scopus)

    Abstract

    We show that if a small holomorphic Sobolev space on the unit disk is not just small but very small, then a trivial necessary condition is also sufficient for a composition operator to be bounded. A similar result for holomorphic Lipschitz spaces is also obtained. These results may be viewed as boundedness analogues of Shapiro's theorem concerning compact composition operators on small spaces. We also prove the converse of Shapiro's theorem if the symbol function is already contained in the space under consideration. In the course of the proofs we characterize the bounded composition operators on the Zygmund class. Also, as a by-product of our arguments, we show that small holomorphic Sobolev spaces are algebras.

    Original languageEnglish
    Pages (from-to)357-380
    Number of pages24
    JournalIntegral Equations and Operator Theory
    Volume56
    Issue number3
    DOIs
    Publication statusPublished - 2006 Nov

    Keywords

    • Composition operator
    • Fractional derivative
    • Small space

    ASJC Scopus subject areas

    • Analysis
    • Algebra and Number Theory

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