Compact double differences of composition operators on the Bergman spaces

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    24 Citations (Scopus)

    Abstract

    As is well known on the weighted Bergman spaces over the unit disk, compactness of differences of two composition operators is characterized by certain cancellation property of the inducing maps at every “bad” boundary point, which makes each composition operator in the difference fail to be compact. Recently, the second and third authors obtained a result implying that double difference cancellation is not possible for linear combinations of three composition operators. In this paper, we obtain a complete characterization for compact double differences formed by four composition operators. Applying our characterization, we easily recover known results on linear combinations of two or three composition operators. As another application, we also show that double difference cancellation is possible for linear combinations of four composition operators by constructing an explicit example of a compact double difference formed by two noncompact differences. In spite of such an example, our characterization also shows that double difference cancellation may occur in the global sense only, and that genuine double difference cancellation is not possible in a certain local sense.

    Original languageEnglish
    Pages (from-to)2273-2307
    Number of pages35
    JournalJournal of Functional Analysis
    Volume272
    Issue number6
    DOIs
    Publication statusPublished - 2017 Mar 15

    Bibliographical note

    Funding Information:
    B.R. Choe was supported by NRF ( 2015R1D1A1A01057685 ) of Korea, H. Koo was supported by NRF ( 2014R1A1A2054145 ) of Korea and NSFC ( 11271293 ), and M. Wang was supported by NSFC ( 11271293 ).

    Publisher Copyright:
    © 2016 Elsevier Inc.

    Copyright:
    Copyright 2017 Elsevier B.V., All rights reserved.

    Keywords

    • Compact operator
    • Composition operator
    • Double difference
    • Weighted Bergman space

    ASJC Scopus subject areas

    • Analysis

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