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IMCS/Publications/BASM/Issues/BASM n.1(92), 2020/

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
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