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

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

2011-09-10

arxiv.org/abs/1109.2220v2

Mathematics

Functional Analysis

Scientific paper

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 $\G:\gH^2\to\HH$ for $A^*$, where $\gH$ is a Hilbert space, $A$ is a symmetric linear relation in $\gH, \cH_0$ is a boundary Hilbert space and $\cH_1$ is a subspace in $\cH_0$. 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 $n_\pm(A)$. 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),\; -\infty\leq a

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