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### 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)
author
publishing date
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
2009-09-16 14:15:10
date last changed
2018-05-29 10:03:22
@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 $H^\infty(\Omega)$, $0&lt;p&lt;\infty$ 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},
}