Sequence of invertible operators converging in norm to an injective but not surjective operator?

Question

Let $X$ be a complex Banach space. Can you find a sequence $A_n$ invertible and $A$ injective but not surjective in $B(X)$ such that $A_n\rightarrow A$ in norm?

I know that such an $A$ must not be bounded below, since then there is a uniform bound on $\Vert A_n^{-1}\Vert$. This implies $A$ would be invertible.

Answer 1

Let $X=L^2[0,\infty)$ and let $Af = e^{-x}f$. $A$ is injective, but not surjective. Let $A_n f = (e^{-x}\chi_{[0,n]}+e^{-n}\chi_{[n,\infty)})f$. Each $A_n$, $n=1,2,3,\cdots$, is invertible, and $$ (A-A_n)f = (e^{-n}-e^{-x})\chi_{[n,\infty)}f, $$ which gives $\|A-A_n\| \le e^{-n}$.