Abstract: Given a densely defined and closed (but not necessarily symmetric) operator A acting on a complex Hilbert space $${\mathcal {H}}$$ , we establish a one-to-one correspondence between its closed extensions and subspaces $${\mathfrak {M}}\subset {\mathcal {D}}(A^*)$$ , that are closed with respect to the graph norm of $$A^*$$ and satisfy certain conditions. In particular, this will allow us to characterize all densely defined and closed restrictions of $$A^*$$ . After this, we will express our results using the language of Gel’fand triples generalizing the well-known results for the selfadjoint case.