On the codimension of the range of a composition operator
Författare
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.
Publiceringsår
1988
Språk
Engelska
Sidor
323-326
Publikation/Tidskrift/Serie
Rendiconti del Seminario Matematico
Volym
46
Issue
3
Länkar
Dokumenttyp
Artikel i tidskrift
Förlag
Seminario Matematico
Ämne
- Mathematics
Status
Published
ISBN/ISSN/Övrigt
- ISSN: 0373-1243