Extension of dissipative operators in a Hilbert space with an indefinite metric

Let $\mathscr H$ be a Hilbert space with an indefinite metric $[x,y]=(\mathscr Jx,y)$ where $\mathscr J$ is an arbitrary unitary and self-adjoint operator in $\mathscr H$ B is a closed $\mathscr J$-dissipative operator in $\mathscr H$ with an arbitrary domain of definition $\mathscr D(B)$ A description is given of all closed (and, in particular, closed maximal) $\mathscr J$-dissipative extensions $\widetilde B$ of the operator $B$ in terms of the corresponding extensions $\widetilde W$ of a nonexpansive operator $W$ in a definite way associated with an initial operator $B$.