On self-adjoint extensions of linear relations generated by integral equations

V. M. Bruk

Taurida Journal of Computer Science Theory and Mathematics, no. 1 (2019), pp. 43-61

Résumé

In the present work, we consider the integral equation \begin{equation*}\label{brukinturZ22041} y(t)=x_{0}-iJ\!\!\int_{[a,t)}\!d\mathbf{p}(s)y(s)-iJ\!\int_{[a,t)}\!d\mathbf{m}(s)f(s), \end{equation*} where $t\in[a,b]$, $b>a$; $y$ is a unknown function; $\mathbf{p}$, $\mathbf{\mathbf{m}}$ 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 $H$, $J=J^{*}$, $J^{2}=E$. We assume that $\mathbf{p}$, $\mathbf{m}$ are measures with bounded variations; $\mathbf{p}$ is a self-adjoint measure; $\mathbf{m}$ is a continuous measure; $x_{0}\in H$; a function $f\in L_{2}(H,d\mathbf{m};a,b)$. We define a minimal relation $L_{0}$ generated by this integral equation and give a description of the adjoint relation $L^{*}_{0}$. We construct a space of boundary values (a boundary triplet) under the condition that the measure $\mathbf{p}$ has single-point atoms $\{t_{k}\}$ such that $t_{k}$ and $t_{k}\!\rightarrow\!b$ as $k\!\rightarrow\!\infty$. We use the obtained results to a description of self-adjoint extensions of the minimal relation $L_{0}$.

Détail

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