Theory of the self-adjoint extensions of symmetric operators with a spectral gap (Q1067568)

Let T be a closed densily defined symmetric operator in the Hilbert space \({\mathcal H}\) with \(\| Tf\| \geq \| f\|\), \(f\in {\mathcal D}(T)\). For the self-adjoint operator A, d(A;\(\Delta)\) denotes the dimension of the subspace \(F_ A(\Delta){\mathcal H}\), where \(\Delta \subset R^ 1\) and \(F_ A(\lambda)\) is the spectral measure of the operator A. For \(p=0,1,2,..\). and \(-1\leq a<b\leq 1\), \({\mathcal F}(a,b;p)\) denotes the following class of operators: \({\mathcal F}(a,b;p)=\{T_{\alpha}:d(T_{\alpha};(a,b))=p\}\), where \(T_{\alpha}\) is the self-adjoint extension of the operator T. The aim of this article is to find all \(T_{\alpha}\in {\mathcal F}(a,b;p)\). The main result is stated as the following: Let \(-1\leq a<b\leq 1\), \(p=0,1,2,..\). \(\lambda_ 0\in (a,b)\), then the equality \[ (T_{\alpha}-\lambda_ 0)^{-1}-(T_{\alpha}-\lambda_ 0)^{- 1}=P(\lambda_ 0)c_{\alpha}P(\lambda_ 0) \] determines interunivalent correlation between the classes \({\mathcal F}(a,b;p)\) and \(\ell_{\lambda_ 0}(a,b;p)\), where \(\ell_{\lambda_ 0}(a,b;p)=\{c_{\alpha}: P(\lambda_ 0){\mathcal H}\to P(\lambda_ 0){\mathcal H}| c_{\alpha}\geq 0\), \(d(c_{\alpha}-f_{a,\lambda_ 0}(b);(0,\infty))=p\}\).