1
$\begingroup$

Fix $N>1$. Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$ be such that the composition operator via $$ \begin{aligned} C_f:C^{\infty}(\mathbb{R},\mathbb{R}) &\rightarrow C^{\infty}(\mathbb{R},\mathbb{R}) \\ g & \mapsto g \circ f, \end{aligned} $$ is a bounded operator. Here the topology on $C^{\infty}(\mathbb{R},\mathbb{R})$ is the Whitney topology.

Let $C^f$ denote the adjoint operator $C_f$. Is there a criterion on $f$ to verify when $C^f$ has at most $N$ linearly independent eigenvectors?

Improve this question
$\endgroup$
4
  • $\begingroup$ What topology do you use on $C(\mathbb{R}, \mathbb{R})$ (so that it makes sense to talk about a bounded operator and about the adjoint operator)? $\endgroup$
    – Jochen Glueck
    Commented Apr 18, 2020 at 19:35
  • 1
    $\begingroup$ @JochenGlueck I meant $C^{\infty}$; also, the topology is Whitney's one. $\endgroup$
    – ABIM
    Commented Apr 19, 2020 at 13:07
  • $\begingroup$ Thank you for the clarification. $\endgroup$
    – Jochen Glueck
    Commented Apr 19, 2020 at 15:26
  • $\begingroup$ My pleasure :) Thanks for pointing it out. $\endgroup$
    – ABIM
    Commented Apr 20, 2020 at 5:17

0

Reset to default

You must log in to answer this question.