Here is a problem I'm working on:
Suppose $A$ is a compact, Hermitian operator with $\ker A=\{0\}$. Prove that there is a sequence $(B_n)$ of bounded operators so that $$ AB_nx\to x\quad\text{and}\quad B_nAx\to x. $$ Moreover, is it possible to choose $(B_n)$ so that $\|B_nA-I\|_\text{op}=0$?
I am not sure where to start on this question. The definition of compact operator that I know is that if $(x_n)$ is a bounded sequence then $(Ax_n)$ contains a convergent subsequence.
It seems the question is asking for $(B_n)$ converging toward the 'inverse' of $A$; how can one find such a $(B_n)$, knowing that $A$ is not necessarily invertible? Any help on this question would be appreciated!
