Eigenvalues for a bounded operator

[Wuberdall]

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!

 

[RUber]

If C were to have any eigenvalues, they would apply to any choice of f in your space.
Consdider a function that is has the property f(0)=0 and f(-1) ##\neq## 0.
Such a function clearly exists in your space.

 

[Wuberdall]

If C were to have any eigenvalues, they would apply to any choice of f in your space.
Consdider a function that is has the property f(0)=0 and f(-1) ##\neq## 0.
Such a function clearly exists in your space.

Thanks for your reply.
I still don't see how this implies that [itex]C[/itex] doesn't have any eigenvalues ?

I mean, does i not only show that there doesn't exist an eigenfunction having these properties?

 

[Wuberdall]

Is this a valid argument ?
Assume that [itex]f:\mathbb{R}\rightarrow\mathbb{R}[/itex] is non vanishing almost everywhere and [itex]\lambda[/itex] is an eigenvalue of [itex]C[/itex], then
[itex]f(2x-1) = \lambda f(x)[/itex]
[itex]2f'(2x-1) = \lambda f'(x)[/itex]

For x=1 the first of the equations yields : [itex]\hspace{3pt} f(1) = \lambda f(1) \hspace{12pt}\Rightarrow\hspace{12pt} \lambda=1[/itex].
While the second of the equations yields : [itex]\hspace{3pt} 2f'(1) = \lambda f'(1) \hspace{12pt}\Rightarrow\hspace{12pt} \lambda=2[/itex].
This is obvious a contradiction, therefore [itex]\lambda[/itex] can't be a eigenvalue.

Of course [itex] f [/itex] must behave properly around [itex] x=1 [/itex], but this is guaranteed by [itex] f \in L_{2}(\mathbb{R}) [/itex].

 

[Ray Vickson]

If C were to have any eigenvalues, they would apply to any choice of f in your space.
Consdider a function that is has the property f(0)=0 and f(-1) ##\neq## 0.
Such a function clearly exists in your space.

No, the eigenvalue equation ##f(2x-1) = \lambda f(x) \; \forall x## need apply only to an eigenfunction ##f = f_{\lambda}##.

 

[RUber]

Thanks, Ray. I must have been operating without caffeine when I posted that.

Wuberdall, your contradiction argument looks reasonable to me.

 

[LCKurtz]

But ##f\in \mathbb L^2## doesn't give you that ##f## is differentiable.

 

[RUber]

LCKurtz makes a good point. The goal should be to contradict your assumptions.
As it stands, 1 is the only potential eigenvalue, as long as f(x) is defined at x=1.
So if f(x) is defined at ##x = 1+ \delta##, ##f(1+ \delta) = f(1+ 2\delta)=f(1+2^k \delta) ## for any delta and integer k .
You may be able to show that this sort of function is either the zero function (trivial) or not in ##L^2##.

 

[Ray Vickson]

Thanks, Ray. I must have been operating without caffeine when I posted that.

Wuberdall, your contradiction argument looks reasonable to me.

More generally: if ##f \in C^{\infty}## near ##x = 1## then we have ##2^n f^{(n)}(2x-1) = \lambda f^{(n)}(x), n = 0,1,2,\ldots##, where ##f^{(n)}## denotes the ##n##th derivative. Thus ##2^n f^{(n)}(1) = \lambda f^{(n)}(1), n = 0,1,2, \ldots##.
(1) If ##\lambda = 0## that would imply that ##f^{(n) = 0, n = 0,1,2, \ldots##, meaning that ##f## would be non-analytic (but still ##C^{\infty}##. That means that ##f## would not equal its own Taylor series. There are functions like that, and our ##f## would have to be one of them.
(2) If ##\lambda \neq 0## then either (i) ##f(0) = 0##; or (ii) ##\lambda = 1##. In case (i) we have ##2 f'(1) = \lambda f'(1), 4 f''(1) = \lambda f''(1), \ldots##, and this allows ##\lambda = 2, f^{(n)}(1) = 0## for ##n \geq 2## or ##\lambda = 4, f^{(n)}(1)=0## for ##n = 1,3,4,\ldots#, etc. Again, ##f## would not be analytic

LCKurtz makes a good point. The goal should be to contradict your assumptions.
As it stands, 1 is the only potential eigenvalue, as long as f(x) is defined at x=1.
So if f(x) is defined at ##x = 1+ \delta##, ##f(1+ \delta) = f(1+ 2\delta)=f(1+2^k \delta) ## for any delta and integer k .
You may be able to show that this sort of function is either the zero function (trivial) or not in ##L^2##.

The equation ##f(1) = \lambda f(1)## implies ##\lambda = 1## only in the case ##f(1) \neq 0##; otherwise, if ##f(1) = 0## there is no restriction on ##\lambda##.