Boundary triples for integral systems on finite intervals

Abstract

Let P, Q, and W be real functions of bounded variation on [0, l], and let W be nondecreasing. The integral system

\( J\overrightarrow{f}(x)-J\overrightarrow{a}=\underset{0}{\overset{x}{\int }}\left(\begin{array}{cc}\uplambda dW- dQ& 0\\ {}0& dP\end{array}\right)\overrightarrow{f}(t),\kern1em J=\left(\begin{array}{cc}0& -1\\ {}1& 0\end{array}\right) \)

on a finite compact interval [0, l] was considered in [6]. The maximal and minimal linear relations A_max and A_min associated with the integral system (0.1) are studied in the Hilbert space L²(W). It is shown that the linear relation A_min is symmetric with deficiency indices n_±(A_min) = 2 and A_max = \( {A}_{min}^{\ast }. \) Boundary triples for A_max are constructed, and the the corresponding Weyl functions are calculated.