Abstract

Let $A_\mathcal{N}$ be the symmetric operator given by the restriction of $A$ to $\mathcal{N}$, where $A$ is a self-adjoint operator on the Hilbert space $\mathcal{H}$ and $\mathcal{N}$ is a linear dense set which is closed with respect to the graph norm on $D$, the operator domain of $A$. We show that any self-adjoint extension $A_\Theta $ of $A_\mathcal{N}$ such that $D\cap D=\mathcal{N}$ can be additively decomposed by the sum $\,A_\Theta \,=\,\bar{A}+T_\Theta $, where both the operators $\bar{A}$ and $T_\Theta $ take values in the strong dual of $D$. The operator $\bar{A}$ is the closed extension of $A$ to the whole $\mathcal{H}$ whereas $T_\Theta $ is explicitly written in terms of a boundary condition depending on $\mathcal{N}$ and on the extension parameter $\Theta $, a self-adjoint operator on an auxiliary Hilbert space isomorphic to the deficiency spaces of $A_\mathcal{N}$. The explicit connection with both Kreĭn’s resolvent formula and von Neumann’s theory of self-adjoint extensions is given