Composition operators on uniform algebras and the pseudohyperbolic metric

P. Galindo, T. W. Gamelin, Mikael Lindström

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9 Citations (Scopus)


Let A be a uniform algebra, and let φ be a self-map of the spectrum MA of A that induces a composition operator Cφ on A. It is shown that the image of MA under some iterate φn of φ is hyperbolically bounded if and only if φ has a finite number of attracting cycles to which the iterates of φ converge. On the other hand, the image of the spectrum of A under φ is not hyperbolically bounded if and only if there is a subspace of A** "almost" isometric to ℓ∞ on which Cφ** is "almost" a,n isometry. A corollary of these characterizations is that if Cφ is weakly compact, and if the spectrum of A is connected, then φ has a unique fixed point, to which the iterates of φ converge. The corresponding theorem for compact composition operators was proved in 1980 by H. Kamowitz [17].
Original languageEnglish
Pages (from-to)1-20
Number of pages20
JournalJournal- Korean Mathematical Society
Issue number1
Publication statusPublished - 2004
MoE publication typeA1 Journal article-refereed


  • uniform algebra
  • composition operator
  • hyperbolically bounded
  • interpolating sequence

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