On abstract self-adjoint boundary conditions (Q2754846)

It is well known that all selfadjoint extensions of a symmetric operator \(A\) with equal deficiency numbers on a Hilbert space \(H\) can be described in terms of abstract boundary conditions. Namely, there exists a triple \((\mathfrak H,\Gamma_1,\Gamma_2)\), where \(\mathfrak H\) is another Hilbert space, \(\Gamma_1,\Gamma_2:\;D(A^*)\to \mathfrak H\), such that NEWLINE\[NEWLINE (A^*f,g)_H-(f,A^*g)_H=(\Gamma_1f,\Gamma_2g)_{\mathfrak H}-(\Gamma_2f,\Gamma_1g)_{\mathfrak H},\quad f,g\in D(A^*), NEWLINE\]NEWLINE and the mapping \(f\mapsto \{ \Gamma_1f,\Gamma_2f\}\) from \(D(A^*)\) to \(\mathfrak H\oplus \mathfrak H\) is surjective. Given such a triple (called a boundary value space) any self-adjoint extension is a restriction of \(A^*\) to vectors \(f\) satisfying the relation NEWLINE\[NEWLINE U(\Gamma_1+i\Gamma_2)f=(\Gamma_1-i\Gamma_2)f, NEWLINE\]NEWLINE where \(U\) is a unitary operator om \(\mathfrak H\). NEWLINENEWLINENEWLINEThe author studies the dependence of the unitary operator \(U\) parametrizing the extensions, on the boundary value space. It is shown that these operator parameters for different boundary value spaces are connected by a fractional-linear transformation associated with a \(J\)-unitary operator matrix. Note, that a similar approach has been used by the reviewer [see \textit{A. N. Kochubei}, Sov. J. Contemp. Math. Anal., Arm. Acad. Sci. 15, No. 3, 52-64 (1980; Zbl 0513.47019)] for describing the connection between characteristic operator-functions corresponding to different boundary value spaces.