On self-adjoint extensions of operators generated by integral equations

In the present work, we prove the Lagrange formula for the integral equation
\begin{equation*} y(t) = y_{0}-iJ \int_{[a,t)} d\mathbf{p}_{1}(s)y(s)-iJ \int_{[a,t)} d\mathbf{q}(s)f(s), \end{equation*}
where $t\in[a,b]$, $b>a$; $y$ is an unknown function; $\mathbf{p}_{1}$, $\mathbf{q}$ 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 $\mathbf{p}_{1}$, $\mathbf{q}$ are measures with a bounded variation and $\mathbf{q}$ is a self-adjoint measure; a function $f$ is integrable with respect to the measure $\mathbf{q}$. The Lagrange formula contains summands that are related to single-point atoms of the measures $\mathbf{p}_{1}$, $\mathbf{q}$.
We use the obtained results to study of linear operators generated by the equation
\begin{equation*} y(t)=x_{0}-iJ \int_{[a,t)} d\mathbf{p}(s)y(s)-iJ \int_{[a,t)} f(s)ds, \end{equation*}
where $\mathbf{p}$ 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 $\mathbf{p}$ 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.