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\begin{document}
\title{Closed range composition operators for one-dimensional smooth symbols}
\author{Adam Przestacki}
\address{
Faculty of Mathematics and Computer Science\\
Adam Mickiewicz
University in Pozna\'{n}, Umultowska 87, 61-614 Pozna\'{n}, Poland\\
}
\email{adamp@amu.edu.pl}
\maketitle
\bigskip
Composition operators are one of the most natural operators acting on the space of smooth functions, which is very important for classical analysis. Deep understanding of their behaviour provides not only interesting information about themselves, but also about the space of smooth functions and its elements.
The aim of the talk is to discuss the following problem: for which smooth symbols $\psi$ the composition operator $C_\psi:C^\infty(\mathbb{R})\to C^\infty(\mathbb{R})$, $F\mapsto F\circ\psi$ has closed range i.e. when the set $\operatorname{Im}$ $C_\psi=\{F\circ\psi:F\circ\psi\}$ is closed in the space of smooth functions endowed with the standard topology of uniform convergence of functions and all derivatives on compact sets. We give a full characterization of such symbols. In particular we prove the following result.
\begin{theorem}
Let $\psi:\mathbb{R}\to\mathbb{R}$ be a smooth function such that there are no points at which all derivatives of $\psi$ vanish. Then the composition operator $C_\psi:C^\infty(\mathbb{R})\to C^\infty(\mathbb{R})$, $F\mapsto F\circ\psi$ has closed range.
\end{theorem}
\bigskip
\begin{thebibliography}{99}
\bibitem{item1} \textsc{Adam Przestacki},
\emph{Composition operators with closed range for one-dimensional smooth symbols},
J. Math. Anal. Appl. \textbf{399} (2013), 225--228.
\bibitem{item1} \textsc{Adam Przestacki},
\emph{Characterization of composition operators with
closed range for one-dimensional smooth symbols },
Preprint
\end{thebibliography}
\end{document}