# Invertibility of linear operators on infinite dimensional vector space

Let $E$ be a finite dimensional vector space, and $T$ a bounded linear operator from $E\rightarrow E$. In this case:

$$T \text{ invertible} \leftrightarrow \text{null } T = \{0\}$$

In an infinite dimensional vector space, this is not the case, but I am struggling to see how that is true. Can you help me find an example of an invertible operator that has a non-trivial nullspace? Or if that does not exist, then an operator with a trivial null space that is not invertible?

For some context, I am trying to better understand the difference between spectrum of an operator and its eigenvalues--and this seems to be the central issue.

Consider the shift operator defined by $T(e_n)=e_{n+1}$ in $Vect(e_0,..,e_n,...)$ its image does not contains $e_0$, so it is not surjective.