On the codimension of the range of a composition operator

Summary, in English

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: Alexandru Aleman
• Publishing year: 1988
• Language: English
• Pages: 323-326
• Publication/Series: Rendiconti del Seminario Matematico
• Volume: 46
• Issue: 3
• Links: http://seminariomatematico.dm.unito.it/rendiconti/cartaceo/46-3/323.pdf
• Document type: Journal article
• Publisher: Seminario Matematico
• Topic: Mathematics
• Status: Published
• ISBN/ISSN/Other: ISSN: 0373-1243