On self-adjoint extensions of operators generated by integral equations.

In the present work, we prove the Lagrange formula for the integral equation
\begin{align}y(t)=y_{0}-iJ\int_{[a,t)}dp_{1}(s)y(s)-iJ\int_{[a,t)}dq(s)f(s),\end{align}
where $t\in [a,b], b> a$; y is an unknown function; $p_{1}, q$ are operator-valued measures defined on Borel sets $\Delta \subset [a,b]$ and taking values in the set of linear bounded operators acting in a separable Hilbert space $H$;$ J$ is a linear operator in $H,J=J^{*},J^{2}=E$. We assume that $p_{1}, q$ are measures with a bounded variation and $q$ is a self-adjoint measure; a function f is integrable with respect to the measure $ q$. The Lagrange formula contains summands that are related to single-point atoms of the measures $ p_{1}, q$.
We use the obtained results to study of linear operators generated by the equation
\begin{align}y(t)=x(0)-iJ\int_{[a,t)}dp(s)y(s)-iJ\int_{[a,t)}f(s)ds,\end{align}
where $p$ is a self-adjoint operator-valued measure with bounded variation; $x_{0}\in H;f\in L_{1}(H;a,b)$. We introduce a minimal symmetric operator generated by this equation and construct a space of boundary values (boundary triplet) under the condition that the measure $p$ has a finite number of single-point atoms. This allows us, with the aid of boundary values, to describe self-adjoint extensions of the symmetric operator generated by the integral equation.
Keywords: Hilbert space, integral equation, operator measure, symmetric operator, self-adjont extension, linear relation, boundary value.