- #1
member 428835
- Homework Statement
- Show no eigenvalues exist for the operator ##Af(x) = xf(x)## where ##A:C[0,1]\to C[0,1]##.
- Relevant Equations
- Nothing comes to mind.
Eigenvalues ##\lambda## for some operator ##A## satisfy ##A f(x) = \lambda f(x)##. Then
$$
Af(x) = \lambda f(x) \implies\\
xf(x) = \lambda f(x)\implies\\
(\lambda-x)f(x) = 0.$$
How do I then show that no eigenvalues exist? Seems obvious one doesn't exists since ##\lambda-x \neq 0## for all ##x\in [0,1]##.
$$
Af(x) = \lambda f(x) \implies\\
xf(x) = \lambda f(x)\implies\\
(\lambda-x)f(x) = 0.$$
How do I then show that no eigenvalues exist? Seems obvious one doesn't exists since ##\lambda-x \neq 0## for all ##x\in [0,1]##.