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 0.1 $$ J\overrightarrow{f}(x)-J\overrightarrow{a}=\underset{0}{\overset{x}{\int }}\left(\begin{array}{cc}\uplambda dW- dQ& 0 {}0& dP\end{array}\right)\overrightarrow{f}(t),\kern1em J=\left(\begin{array}{cc}0& -1 {}1& 0\end{array}\right) $$ 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.