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## Homework Statement

Let [itex]C[/itex] be the composition operator on the Hilbert space [itex]L_{2}(\mathbb{R})[/itex] with the usual inner product. Let [itex]f\in L_{2}(\mathbb{R})[/itex], then [itex]C[/itex] is defined by

[itex](Cf)(x) = f(2x-1)[/itex], [itex]\hspace{9pt}x\in\mathbb{R}[/itex]

give a demonstration, which shows that [itex]C[/itex] does not have any eigenvalues.

## Homework Equations

[itex]C[/itex] is a unitary operator.

Let [itex]\mathcal{F}[/itex] denote the Fourier Transformation on [itex]L_{2}(\mathbb{R})[/itex], then

[itex](\mathcal{F}C\mathcal{F}^{\ast}f)(p) =

\frac{\exp\big(-i\tfrac{1}{2}p\big)}{\sqrt{2}}\hspace{1pt}f\big(\tfrac{1}{2}p\big)

[/itex]

## The Attempt at a Solution

Direct application of the eigenvalue equation of course yields Schröders equation, that is

[itex]f(2x-1) = \lambda f(x)[/itex]

I don't have a slightest idea on how to proceed from here.

Any good suggestions are more than welcome!