| Title | On the codimension of the range of a composition operator |
| Authors | Alexandru Aleman |
| Alternative Location | http://seminariomatematico.... |
| Publication | Rendiconti del Seminario matematico |
| Year | 1988 |
| Volume | 46 |
| Issue | 3 |
| Pages | 323 - 326 |
| Document type | Article |
| Status | Published |
| Quality controlled | Yes |
| Language | eng |
| Publisher | Seminario Matematico |
| Abstract 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. |
| ISBN/ISSN/Other | ISSN: 03731243 |
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