Given a nilpotent operator $L$ in a certain space, can we always find another space such that $L$ is not nilpotent?

Question

I just thought the following (perhaps silly) observation: If we have D, the differentiation operator in the space of polynomials of $\deg \leq n$, then $D$ is nilpotent. But if we consider a space such as the space spanned by $\{ e^x\}$, then $D$ won't be nilpotent. So being nilpotent depends not only on the operator, it also depends on the space it acts.

• Given a nilpotent operator $L$ in a certain space, can we always find another space such that $L$ is not nilpotent?

• Given a non-nilpotent operator $L$ in a certain space, can we always find another space in which the $L$ is nilpotent?