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Compact composition operators and iteration

Aleman, Alexandru LU (1993) In Journal of Mathematical Analysis and Applications 173(2). p.550-556
Abstract
Let $\Omega$ be a domain in the complex plane, $\phi$ an analytic map that maps $\Omega$ into itself, and $X$ an $F$-space of analytic functions in $\Omega$ that possesses certain mild regularity properties. (Some examples considered in the paper are Hardy spaces, Bergman spaces, and the space of all analytic functions in $\Omega$.) If composition with $\phi$ defines a compact operator on $X$ that has eigenvalues, then the iterates of $\phi$ converge to a constant $\lambda$ in the closure of $\Omega$. Furthermore, if $\lambda\in\partial\Omega$ and $\liminf_{\zeta\to\lambda}|\phi(\zeta)-\lambda|/|\zeta-\lambda| > 0$, then functions in $H^\infty(\Omega)$ and their derivatives have nice behavior near $\lambda$ in the sense that the... (More)
Let $\Omega$ be a domain in the complex plane, $\phi$ an analytic map that maps $\Omega$ into itself, and $X$ an $F$-space of analytic functions in $\Omega$ that possesses certain mild regularity properties. (Some examples considered in the paper are Hardy spaces, Bergman spaces, and the space of all analytic functions in $\Omega$.) If composition with $\phi$ defines a compact operator on $X$ that has eigenvalues, then the iterates of $\phi$ converge to a constant $\lambda$ in the closure of $\Omega$. Furthermore, if $\lambda\in\partial\Omega$ and $\liminf_{\zeta\to\lambda}|\phi(\zeta)-\lambda|/|\zeta-\lambda| > 0$, then functions in $H^\infty(\Omega)$ and their derivatives have nice behavior near $\lambda$ in the sense that the functionals of evaluation at $\lambda$ for functions and their derivatives have weak${}^*$-continuous extensions to $H^\infty(\Omega)$.



The following example is discussed in the above context. Start with an analytic map $\phi$ from a disk into itself that has an attractive fixed point $\lambda$ at the center. Form $\Omega$ by removing from the disk both $\lambda$ and a carefully chosen sequence of disjoint disks that converge to $\lambda$. Finally, let $X=H^\infty(\Omega)$. (Less)
Please use this url to cite or link to this publication:
author
publishing date
type
Contribution to journal
publication status
published
subject
in
Journal of Mathematical Analysis and Applications
volume
173
issue
2
pages
550 - 556
publisher
Elsevier
external identifiers
  • Scopus:43949167441
ISSN
0022-247X
language
English
LU publication?
no
id
904aca9d-8e48-4fda-bde7-d6736a7104e1 (old id 1467364)
alternative location
http://ida.lub.lu.se/cgi-bin/elsevier_local?YYUM0070-A-0022247X-V0173I02-83710875
date added to LUP
2009-09-16 13:38:28
date last changed
2016-10-13 04:33:14
@misc{904aca9d-8e48-4fda-bde7-d6736a7104e1,
  abstract     = {Let $\Omega$ be a domain in the complex plane, $\phi$ an analytic map that maps $\Omega$ into itself, and $X$ an $F$-space of analytic functions in $\Omega$ that possesses certain mild regularity properties. (Some examples considered in the paper are Hardy spaces, Bergman spaces, and the space of all analytic functions in $\Omega$.) If composition with $\phi$ defines a compact operator on $X$ that has eigenvalues, then the iterates of $\phi$ converge to a constant $\lambda$ in the closure of $\Omega$. Furthermore, if $\lambda\in\partial\Omega$ and $\liminf_{\zeta\to\lambda}|\phi(\zeta)-\lambda|/|\zeta-\lambda| &gt; 0$, then functions in $H^\infty(\Omega)$ and their derivatives have nice behavior near $\lambda$ in the sense that the functionals of evaluation at $\lambda$ for functions and their derivatives have weak${}^*$-continuous extensions to $H^\infty(\Omega)$. <br/><br>
<br/><br>
The following example is discussed in the above context. Start with an analytic map $\phi$ from a disk into itself that has an attractive fixed point $\lambda$ at the center. Form $\Omega$ by removing from the disk both $\lambda$ and a carefully chosen sequence of disjoint disks that converge to $\lambda$. Finally, let $X=H^\infty(\Omega)$.},
  author       = {Aleman, Alexandru},
  issn         = {0022-247X},
  language     = {eng},
  number       = {2},
  pages        = {550--556},
  publisher    = {ARRAY(0xa265f68)},
  series       = {Journal of Mathematical Analysis and Applications},
  title        = {Compact composition operators and iteration},
  volume       = {173},
  year         = {1993},
}