Infinitely many nonhomogeneous conditions
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.
deficiency index problem for formally self-adjoint second order systems of ordinary differential expression
scientific article; zbMATH DE number 3913059