On self-adjoint extensions of operators generated by integral equations
V. M. Bruk Saratov State Technical University
Abstract:
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.
Keywords:
Hilbert space, integral equation, operator measure, symmetric operator, self-adjontextension, linear relation, boundary value.
UDC:
517.983
MSC: 47A06,
47A10,
34B27
Fulltext:
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