Definition 11

Let $u$ be a closed symmetric operator on a complex Hilbert space E. A boundary condition for $u$ is a partially isometric linear mapping (TVS, V, p. 41, Def. 3) of Ker($u^*-i$) into Ker($u^*+i$).

Let $u$ be a closed symmetric operator on a complex Hilbert space E and $b$ a boundary condition for $u$. Let I = Ker($b$)$^{\circ}$ be the initial subspace of $b$. We denote by $u_b$ the reduction of $u^*$ to the subspace of dom($u^*$) which is the direct sum of dom($u$) and of the graph in $E_+\oplus E_-$ of the restriction of $b$ to I. Since $E_{u^*}=$ dom($u$)$\oplus E_+\oplus E_-$ (Prop. 26), the partial operator $u_b$ is well defined.

In other words, the domain of $u_b$ is the space of elements $x\in E$ of the form $x=x_0+y+b(y)$, where $x_0\in$ dom($u$) and $y$ belongs to the initial subspace of $b$. One then has $u_b(x_0+y+b(y)) =u(x_0) +iy-ib(y)$.