Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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COMPOSITION OPERATORS ON UNIFORM ALGEBRAS AND THE PSEUDOHYPERBOLIC METRIC

  • Galindo, P. (Departamento de Analisis Matematico Universidad de Valencia) ;
  • Gamelin, T.W. (Department of Mathematics UCLA) ;
  • Lindstrom, M. (Department of Mathematics Abo Akademi University)
  • Published : 2004.01.01
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Abstract

Let A be a uniform algebra, and let $\phi$ be a self-map of the spectrum $M_A$ of A that induces a composition operator $C_{\phi}$, on A. It is shown that the image of $M_A$ under some iterate ${\phi}^n$ of \phi is hyperbolically bounded if and only if \phi has a finite number of attracting cycles to which the iterates of $\phi$ converge. On the other hand, the image of the spectrum of A under $\phi$ is not hyperbolically bounded if and only if there is a subspace of $A^{**}$ "almost" isometric to ${\ell}_{\infty}$ on which ${C_{\phi}}^{**}$ "almost" an isometry. A corollary of these characterizations is that if $C_{\phi}$ is weakly compact, and if the spectrum of A is connected, then $\phi$ has a unique fixed point, to which the iterates of $\phi$ converge. The corresponding theorem for compact composition operators was proved in 1980 by H. Kamowitz [17].

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  3. Quasicompact and Riesz endomorphisms of Banach algebras vol.225, pp.2, 2005, https://doi.org/10.1016/j.jfa.2005.04.002