On $C$-Symmetric and $C$-Self-adjoint Unbounded Operators on Hilbert Space

Let $C$ be a conjugation on a Hilbert space $\mathcal{H}$. A densely defined linear operator $A$ on $\mathcal{H}$ is called $C$-symmetric if $CAC\subseteq A^{\ast }$ and $C$-self-adjoint if $CAC=A^{\ast }$. Our main results describe all $C$-self-adjoint extensions of $A$ on $\mathcal{H}$. Further, we prove a $C$-self-adjointness criterion based on quasi-analytic vectors and we characterize $C$-self-adjoint operators in terms of their polar decompositions.