Boundary relations and boundary conditions for general (not necessarily definite) canonical systems with possibly unequal deficiency indices

2011/09/10 by Mogilevskii, Vadim
#34A30 #34B40 #47A06 #47B25 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1109.2220

Abstract

We investigate in the paper general (not necessarily definite) canonical systems of differential equation in the framework of extension theory of symmetric linear relations. For this aim we first introduce the new notion of a boundary relation \G:\gH²→\HH for A^*, where \gH is a Hilbert space, A is a symmetric linear relation in \gH, \cH₀ is a boundary Hilbert space and \cH₁ is a subspace in \cH₀. Unlike known concept of a boundary relation (boundary triplet) for A^* our definition of \G is applicable to relations A with possibly unequal deficiency indices n_±(A). Next we develop the known results on minimal and maximal relations induced by the general canonical system J y'(t)-B(t)y(t)=\D (t)f(t) on an interval \cI=(a,b), -∞≤ a<b compact></b>

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