Extensions of symmetric operators with distinct deficiency numbers
The present paper builds upon the development in \textit{A. N. Kochubej}, Mat. Zametki 17, No.1, 41-48 (1975; Zbl 0322.47006) and as in that paper, describes extensions in terms of ''abstract boundary conditions.'' Let \(L_ 0\) be a symmetric operator, \(L=L^*_ 0\), and \(H_ 0^{\pm}=\ker nel\) (L\(\mp iI)\). A quadruple \((H^+,H^- ,\delta_+,\delta_-)\) is an abstract space of boundary values (a.s.b.v) of \(L_ 0\) if \(\delta =\delta_++\delta_ 1\), dim \(H^{\pm}=\dim H_ 0^{\pm}\), \(R(\delta)=H^+\oplus H^-\), \(\ker nel(\delta)=D(L_ 0)\) and \[ <Ly,z>-<y,Lz>=i(<\delta_+y,\delta_+z>- <\delta_-y,\delta_-z>) \] (Green's formula). The fundamental result of the paper is (in part) \(L_ 1\) is a maximal dissipative extension of \(L_ 0\) if and only if there is an operator \(K: H^+\to H^-\) such that \(D(K)=H^+\), \(\| K\| \leq 1\) and \[ D(L_ 1)=\{y\in D(L):\quad \delta_-y=K\delta_+y\}. \]
scientific article; zbMATH DE number 3958044