On the codimension of the range of a composition operator
(1988) In Rendiconti del Seminario Matematico 46(3). p.323-326- Abstract
- Let $\Omega$ be a domain in ${\bf C}$. A point $\lambda$ of the boundary of $\Omega$ is said to be essential if, for every neighborhood $V$ of $\lambda$, there is an $f\in H^\infty(\Omega)$ such that $f$ does not extend analytically to $V$. It is known that there is a smallest domain $\Omega^*$ containing $\Omega$ such that $\Omega^*$ has no nonessential boundary points. The main result here is the following: Suppose $H^\infty(\Omega)$ is nontrivial. Let $\varphi\colon\Omega\to\Omega$ be analytic and let $C_\varphi$ be the bounded linear operator on $H^\infty(\Omega)$, $0<p<\infty$ given by $C_\varphi f=f\circ\varphi$. Then the range of $C_\varphi$ has uncountable codimension unless $\varphi$ extends to a conformal mapping of... (More)
- Let $\Omega$ be a domain in ${\bf C}$. A point $\lambda$ of the boundary of $\Omega$ is said to be essential if, for every neighborhood $V$ of $\lambda$, there is an $f\in H^\infty(\Omega)$ such that $f$ does not extend analytically to $V$. It is known that there is a smallest domain $\Omega^*$ containing $\Omega$ such that $\Omega^*$ has no nonessential boundary points. The main result here is the following: Suppose $H^\infty(\Omega)$ is nontrivial. Let $\varphi\colon\Omega\to\Omega$ be analytic and let $C_\varphi$ be the bounded linear operator on $H^\infty(\Omega)$, $0<p<\infty$ given by $C_\varphi f=f\circ\varphi$. Then the range of $C_\varphi$ has uncountable codimension unless $\varphi$ extends to a conformal mapping of $\Omega^*$ onto itself. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/1467383
- author
- Aleman, Alexandru ^{LU}
- publishing date
- 1988
- type
- Contribution to journal
- publication status
- published
- subject
- in
- Rendiconti del Seminario Matematico
- volume
- 46
- issue
- 3
- pages
- 323 - 326
- publisher
- Seminario Matematico
- ISSN
- 0373-1243
- language
- English
- LU publication?
- no
- id
- 7a81240f-3f8f-4078-a11c-af5d7a552662 (old id 1467383)
- alternative location
- http://seminariomatematico.dm.unito.it/rendiconti/cartaceo/46-3/323.pdf
- date added to LUP
- 2009-09-16 14:15:10
- date last changed
- 2016-06-29 09:18:37
@article{7a81240f-3f8f-4078-a11c-af5d7a552662, abstract = {Let $\Omega$ be a domain in ${\bf C}$. A point $\lambda$ of the boundary of $\Omega$ is said to be essential if, for every neighborhood $V$ of $\lambda$, there is an $f\in H^\infty(\Omega)$ such that $f$ does not extend analytically to $V$. It is known that there is a smallest domain $\Omega^*$ containing $\Omega$ such that $\Omega^*$ has no nonessential boundary points. The main result here is the following: Suppose $H^\infty(\Omega)$ is nontrivial. Let $\varphi\colon\Omega\to\Omega$ be analytic and let $C_\varphi$ be the bounded linear operator on $H^\infty(\Omega)$, $0<p<\infty$ given by $C_\varphi f=f\circ\varphi$. Then the range of $C_\varphi$ has uncountable codimension unless $\varphi$ extends to a conformal mapping of $\Omega^*$ onto itself.}, author = {Aleman, Alexandru}, issn = {0373-1243}, language = {eng}, number = {3}, pages = {323--326}, publisher = {Seminario Matematico}, series = {Rendiconti del Seminario Matematico}, title = {On the codimension of the range of a composition operator}, volume = {46}, year = {1988}, }