RO  EN

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

Authors: V. M. Bruk

Abstract

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 $$\theta$$ ente\-ring 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 rela\-tions $$L_{\theta(\lambda)}$$ and $$\theta(\lambda)$$ are holomorphic together.

Saratov State Technical University
77, Politehnicheskaja str., Saratov 410054
Russia
E-mail:

0.20 Mb