Im reading a script where there is stated that there is a bijection from the set of all operators $A: D(A)\subset H\rightarrow H$ ($A=\overline{A}\subset A^*$) to the set of all operators $V: D(V)\subset H\rightarrow H$ where $D(V)$ is closed and $\mathrm{ran}(I-V)$ is dense in $H$. Where $H$ is a Hilbert space.
The proof is done by showing that for a given $V$ like above, the operator $A:=i(I+V)(I-V)^{-1}$ is closed and symmetric. Where $D(A):=\mathrm{ran}(I-V)$.
The injectiviy I guess should follow by the Cayley transform of $A$.
Where I stuck particullary is to show that $A$ is well defined. Therefore why does $(I-V)^{-1}$ exist or why is $\mathrm{ker}(I-V)=\lbrace 0\rbrace$?