Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 1, pp. 1-20.

Let ${A}_{𝒩}$ be the symmetric operator given by the restriction of $A$ to $𝒩$, where $A$ is a self-adjoint operator on the Hilbert space $ℋ$ and $𝒩$ is a linear dense set which is closed with respect to the graph norm on $D\left(A\right)$, the operator domain of $A$. We show that any self-adjoint extension ${A}_{\Theta }$ of ${A}_{𝒩}$ such that $D\left({A}_{\Theta }\right)\cap D\left(A\right)=𝒩$ can be additively decomposed by the sum $\phantom{\rule{0.166667em}{0ex}}{A}_{\Theta }\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\overline{A}+{T}_{\Theta }$, where both the operators $\overline{A}$ and ${T}_{\Theta }$ take values in the strong dual of $D\left(A\right)$. The operator $\overline{A}$ is the closed extension of $A$ to the whole $ℋ$ whereas ${T}_{\Theta }$ is explicitly written in terms of a (abstract) boundary condition depending on $𝒩$ and on the extension parameter $\Theta$, a self-adjoint operator on an auxiliary Hilbert space isomorphic (as a set) to the deficiency spaces of ${A}_{𝒩}$. The explicit connection with both Kreĭn’s resolvent formula and von Neumann’s theory of self-adjoint extensions is given.

Classification : 47B25,  47B38,  47F05
@article{ASNSP_2003_5_2_1_1_0,
author = {Posilicano, Andrea},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {1--20},
publisher = {Scuola normale superiore},
volume = {Ser. 5, 2},
number = {1},
year = {2003},
zbl = {1096.47505},
mrnumber = {1990972},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2003_5_2_1_1_0/}
}
Posilicano, Andrea. Self-adjoint extensions by additive perturbations. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 1, pp. 1-20. http://www.numdam.org/item/ASNSP_2003_5_2_1_1_0/

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