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Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, 2020, номер 1, страницы 106–121
(Mi basm526)
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Эта публикация цитируется в 1 научной статье (всего в 1 статье)
Research articles
On self-adjoint and invertible linear relations generated by integral equations
V. M. Bruk Saratov State Technical University 77, Politehnicheskaja str., Saratov 410054 Russia
Аннотация:
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.
Ключевые слова и фразы:
integral equation, Hilbert space, boundary value problem, operator measure, linear relation, spectrum.
Поступила в редакцию: 17.03.2020
Образец цитирования:
V. M. Bruk, “On self-adjoint and invertible linear relations generated by integral equations”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2020, no. 1, 106–121
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/basm526 https://www.mathnet.ru/rus/basm/y2020/i1/p106
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Страница аннотации: | 195 | PDF полного текста: | 60 | Список литературы: | 16 |
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