Geodesic
Extensions and generalized resolvents of a symmetric operator which is not densely defined
Izvestiya. Mathematics , Tome 4 (1970) no. 1, pp. 179-208

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We consider extensions of a closed symmetric operator $A$ whose domain is, in general, not dense in the given Hilbert space $H$. In particular, we study self-adjoint extensions outside $H$ and the one-parameter families of operators $B_\lambda\supset A$ ($\operatorname{Im}\lambda\ne0$) generated by them in $H$ which are dissipative for $\operatorname{Im}\lambda0$. The set of all generalized resolvents of the operator $A$ is characterized. 
@article{IM2_1970_4_1_a9,
     author = {A. V. Strauss},
     title = {Extensions and generalized resolvents of a symmetric operator which is not densely defined},
     journal = {Izvestiya. Mathematics },
     pages = {179--208},
     publisher = {mathdoc},
     volume = {4},
     number = {1},
     year = {1970},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1970_4_1_a9/}
}
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A. V. Strauss. Extensions and generalized resolvents of a symmetric operator which is not densely defined. Izvestiya. Mathematics , Tome 4 (1970) no. 1, pp. 179-208. http://geodesic.mathdoc.fr/item/IM2_1970_4_1_a9/