Infinitely many nonhomogeneous conditions
type: Item
instance of: scholarly article
title: Infinitely many nonhomogeneous conditions
zbMATH Open document ID: 0572.47004
DOI: 10.1016/0022-247X(85)90050-2
author: B. George
published in: Journal of Mathematical Analysis and Applications
publication date: 1985-01-01T00:00:00Z
review text: Let \(T_ 0\subset T_ 1\) be closed linear relations contained in the direct Hilbert space sum \(H_ 1\oplus H_ 2\) such that \(T_ 1\Theta T_ 0\) is isomorphic to an infinite dimensional separable Hilbert space. Let T be an arbitrary closed linear relation between \(T_ 0\) and \(T_ 1\). The main part of this paper consists of: (1) A relation is found which connects the dimensions of the null spaces of \(T_ 1\), \(T^*_ 0\), T, \(T^*\) with m and \(\tilde m,\) where m and \(\tilde m\) are the numbers (possibly infinite) of linearly independent boundary conditions defining T and \(T^*\), respectively. (2) For a given g, a necessary and sufficient condition for the relation \(\{\) x,g\(\}\in T\) to have a solution. This is expressed in terms of \(T^*\) and the complementary boundary conditions for \(T^*\) without the range condition on T. Finally, the part (2) is applied to the deficiency index problem for formally self-adjoint second order systems of ordinary differential expression.
Mathematics Subject Classification ID: 47A50
Mathematics Subject Classification ID: 34L99
zbMATH DE Number: 3913059
zbMATH Keywords: closed linear relation
zbMATH Keywords: dimensions of the null spaces
zbMATH Keywords: range condition
zbMATH Keywords: deficiency index problem for formally self-adjoint second order systems of ordinary differential expression
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3913059