On self-adjoint and invertible linear relations generated by integral equations
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517.968.2+517.983 (1)
Дифференциальные, интегральные и другие функциональные уравнения. Конечные разности. Вариационное исчисление. Функциональный анализ (243)
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BRUK, Vladislav. On self-adjoint and invertible linear relations generated by integral equations. In: Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, 2020, nr. 1(92), pp. 106-121. ISSN 1024-7696.
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Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica
Numărul 1(92) / 2020 / ISSN 1024-7696 /ISSNe 2587-4322

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

CZU: 517.968.2+517.983
MSC 2010: 46G12, 45N05, 47A10.

Pag. 106-121

Bruk Vladislav
 
Yuri Gagarin State Technical University of Saratov
 
 
Disponibil în IBN: 5 septembrie 2020


Rezumat

We define a minimal operator L0 generated by an integral equation with an operator measure and prove necessary and sufficient conditions for the operator L0 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 bL with the property L0 ⊂ bL⊂L 0 and relations  entering in boundary conditions. In this case we denote bL = L. We establish conditions under which linear relations L and  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 finitedimensional 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 and prove that families of linear relations L() and () are holomorphic together.

Cuvinte-cheie
integral equation, Hilbert space, boundary value problem, operator measure, linear relation, spectrum

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<dc:creator>Bruk, V.M.</dc:creator>
<dc:date>2020-08-20</dc:date>
<dc:description xml:lang='en'><p>We define a minimal operator L0 generated by an integral equation with an operator measure and prove necessary and sufficient conditions for the operator L0 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 bL with the property L0 &sub; bL&sub;L 0 and relations  entering in boundary conditions. In this case we denote bL = L. We establish conditions under which linear relations L and  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 finitedimensional 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 and prove that families of linear relations L() and () are holomorphic together.</p></dc:description>
<dc:source>Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica 92 (1) 106-121</dc:source>
<dc:subject>integral equation</dc:subject>
<dc:subject>Hilbert space</dc:subject>
<dc:subject>boundary value problem</dc:subject>
<dc:subject>operator measure</dc:subject>
<dc:subject>linear relation</dc:subject>
<dc:subject>spectrum</dc:subject>
<dc:title>On self-adjoint and invertible linear relations generated by integral equations</dc:title>
<dc:type>info:eu-repo/semantics/article</dc:type>
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