Let $A: D(A) \subseteq H \rightarrow H$ be a symmetric and closed operator on a Hilbert space $H$.
Denote by $\mathscr{K}_{\pm}:=\mathrm{ran}(A\pm i)^\perp$ and $n_\pm:=\mathrm{dim}(\mathscr{K}_\pm)$ the deficiency indices.
Two things I try to find out
1.) If $n_+=n_-$ then $A$ has a selfadjoint extention $B$. If $U:\mathscr{K}_-\rightarrow \mathscr{K}_+$ is unitary how can I show that $(B+i)(B-i)^{-1}\upharpoonright_{\mathscr{K}_-}=U$?
2.) How can I show $n_+=n_-$ if and only if $A$ has a self adjoint extention?
Regarding 1.) As I know if $A\subseteq B=B^*$ then $C_B:=(B+i)(B-i)^{-1} \in L(H)$ is unitary and $C_B\upharpoonright_{\mathscr{K}_-}: \mathscr{K_-}\rightarrow \mathscr{K_+}$ is a unitary operator.
So 1.) is the converse of the above statement: "If $A\subseteq B=B^*$ and $U: \mathscr{K_-}\rightarrow \mathscr{K_+}$ is unitary. Do we have $(B+i)(B-i)^{-1} \upharpoonright_{\mathscr{K}_-}=U$?"
What I can provide as a tool maybe is:
If $V:D(V)\subset H\rightarrow H$ is an isometrie, with dense range and closed domain. Then $T:=i(I+V)(I-V)^{-1}$ defines a closed, symmetric operator with domain $D(T):=\mathrm{ran}(I-V)$.