Compact composition operators and iteration
(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:
https://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
- 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
- 2016-04-04 09:34:51
- date last changed
- 2021-01-03 09:08:17
@article{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 $\liminf_{\zeta\to\lambda}|\phi(\zeta)-\lambda|/|\zeta-\lambda| > 0$, 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 = {{Elsevier}}, series = {{Journal of Mathematical Analysis and Applications}}, title = {{Compact composition operators and iteration}}, url = {{http://ida.lub.lu.se/cgi-bin/elsevier_local?YYUM0070-A-0022247X-V0173I02-83710875}}, volume = {{173}}, year = {{1993}}, }