Extensions and generalized resolvents of closed dissipative operators

Let $B$ be a closed dissipative operator in a Hilbert space $\mathscr H$ with an arbitrary domain of definition. We investigate briefly the problem of describing all the closed (and, in particular, the closed maximal) dissipative extensions $\widetilde B$ of the operator $B$. Following this we introduce the concept of a generalized resolvent of a closed dissipative operator with an arbitrary domain of definition, and we study the fundamental properties of generalized resolvents.