О самосопряженных расширениях линейных отношений, порожденных\ интегральными\ уравнениями


In the present work, we consider the integral equation
$$y(t) = x_0 - iJ\int_{[a, t)} d\textbf{p}(s)y(s) - iJ\int_{[a, t)}d\textbf{m}(s)f(s)$$
where $t \in [a, b], b > a; y$ is a unknown function; $\textbf{p}, \textbf{m}$ 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
$H, J = J^\ast, J^2 = E$. We assume that $\textbf{p}$, mare measures with bounded variations; $\textbf{p}$ is a self-adjoint measure; $\textbf{m}$ is a continuous measure;$x_0 \in H$; a function $f \in L^2(H, d\textbf{m}; a, b)$. We define a minimal relation $L_0$ generated by this integral equation and give a description of the adjoint relation $L^\ast_0$. We construct a space of boundary values (a boundary triplet) under the condition that the measure $\textbf{p}$ has single-point atoms $\{t_k\}$ such that $t_k < t_{k+1}$ and $t_k \to b$ as $k \to \infty$. We use the obtained results to a description of self-adjoint extensions of the minimal relation $L_0$.

Keywords: Hilbert space, integral equation, operator measure, linear relation, symmetric relation, self-adjoint extension, boundary value.