Compact composition operators and iteration
(1993) In Journal of Mathematical Analysis and Applications 173(2). p.550556 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:
http://lup.lub.lu.se/record/1467364
 author
 Aleman, Alexandru ^{LU}
 publishing date
 1993
 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
 0022247X
 language
 English
 LU publication?
 no
 id
 904aca9d8e484fdabde7d6736a7104e1 (old id 1467364)
 alternative location
 http://ida.lub.lu.se/cgibin/elsevier_local?YYUM0070A0022247XV0173I0283710875
 date added to LUP
 20090916 13:38:28
 date last changed
 20180107 10:24:04
@article{904aca9d8e484fdabde7d6736a7104e1, 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 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 = {0022247X}, language = {eng}, number = {2}, pages = {550556}, publisher = {Elsevier}, series = {Journal of Mathematical Analysis and Applications}, title = {Compact composition operators and iteration}, volume = {173}, year = {1993}, }