On self-adjoint and invertible linear relations generated by integral equations

V. M. Bruk

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2020), pp. 106-121

Résumé

We define a minimal operator $L_{0}$ generated by an integral equation with an operator measure and prove necessary and sufficient conditions for the operator $L_{0}$ to be densely defined. In general, $L^{*}_{0}$ is a linear relation. We give a description of $L^{*}_{0}$ and establish that there exists a one-to-one correspondence between relations $\widehat{L}$ with the property $L_{0} \subset\widehat{ L} \subset L^{*}_{0}$ and relations $\theta$ entering in boundary conditions. In this case we denote $\widehat{L}=L_{\theta}$. We establish conditions under which linear relations $L_{\theta}$ and $\theta$ together have the following properties: a linear relation $(l.r)$ is self-adjoint; $l.r$ is closed; $l.r$ is invertible, i.e., the inverse relation is an operator; $l.r$ has the finite-dimensional kernel; $l.r$ is well-defined; the range of $l.r$ is closed; the range of $l.r$ is a closed subspace of the finite codimension; the range of $l.r$ coincides with the space wholly; $l.r$ is continuously invertible. We describe the spectrum of $L_{\theta}$ and prove that families of linear relations $L_{\theta(\lambda)}$ and $\theta(\lambda)$ are holomorphic together.