The following is an exercise from Halmos book "A Hilbert space problem book" :
Exercise: If $H$ and $K$ are Hilbert spaces, and if $A$ is a bounded linear transformation that maps $H$ one to one and onto $K$, then $A$ is invertible.
He gives the following solution for this:
I do not know why he consider the operator $0$ in the role of $A^*$. Clearly $0$ is not bijective, so why does he state this part? Please help me. Thanks.