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, $L_0^{\ast }$ is a linear relation. We give a description of $L_0^{\ast }$ and establish that there exists a one-to-one correspondence between relations $\hat{L}$ with the property $L_0\subset \hat{L}\subset L_0^{\ast }$ and relations $\theta$ entering in boundary conditions. In this case we denote $\hat{L}=L_{\theta }$. 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.

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