Linear Fractional Composition Operators Over the Half-Plane

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Abstract

In the setting of the Hardy spaces or the standard weighted Bergman spaces over the unit ball in Cn, linear fractional composition operators are known to behave quite rigidly in the sense that they cannot form any nontrivial compact differences or, more generally, linear combinations. In this paper, in the setting of the standard weighted Bergman spaces over the half-plane, we completely characterize bounded/compact differences of linear fractional composition operators. Our characterization reveals that a linear fractional composition operator can possibly form a compact difference, which is a new half-plane phenomenon due to the half-plane not being bounded. Also, we obtain necessary conditions and sufficient conditions for a linear combination to be bounded/compact. As a consequence, when the weights and exponents of the weighted Bergman spaces are restricted to a certain range, we obtain a characterization for a linear combination to be bounded/compact. Applying our results, we provide an example showing a double difference cancellation phenomenon for linear combinations of three linear fractional composition operators, which is yet another half-plane phenomenon.

Original languageEnglish
Article number27
JournalIntegral Equations and Operator Theory
Volume90
Issue number3
DOIs
Publication statusPublished - 2018 Jun 1

Keywords

  • Bounded operator
  • Compact operator
  • Difference
  • Half-plane
  • Hilbert–Schmidt operator
  • Linear combination
  • Linear fractional composition operator
  • Weighted Bergman space

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory

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