Let $$D:=\frac{d}{dx}:C^\infty(\mathbb R,\mathbb C)\to C^\infty(\mathbb R,\mathbb C)$$ be the differential operator. I am interested in
- the Kernel of $D$.
- the Eigenvalues of $D$.
- the Eigenspace of $D$.
For 1. and 2. I just solved the differential equations $f'=0$ and $f'-\lambda f=0$ and obtained $\ker(D)=\mathbb R$ and $f(x)=\exp(\lambda x+c)$. Is it correct to conclude that every element of $\mathbb R$ is an eigenvalue of $D$ with infinitely many eigenvectors $\exp(\lambda x+c)$? Also, clearly $-\exp(-\lambda x+c)$ is also an eigenvector of $D$ for the same eigenvalue and since the Domain of $D$ is $C^\infty(\mathbb R,\mathbb C)$ I can't write $-1$ as an $e^a$ to absorb it in the coefficent $c$. I hope some of you can enlighten me here a bit as I am very confused.
For 2. if $ \lambda$ is an eigenvalue of $D$, then there is $f \in C^\infty(\mathbb R,\mathbb C)$ such that $f \ne 0$ and $f'= \lambda f$. Hence there is $c \ne 0$ such that $f(x)=c e^{\lambda x}$.
Conclusion: each $ \lambda \in \mathbb C$ is an eigenvalue of $D$ and the coresponding eigenspace is
$$ \{ c e^{ \lambda x}: c \in \mathbb C\}.$$