Defect indices and definiteness conditions for a class of discrete linear Hamiltonian systems

Deficiency indices and spectrum of fourth order difference equations with unbounded coefficients

“…It follows that \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$H_{00} \subset \tilde{H} \subset H_{00}^{\ast }$\end{document} and consequently H 00 is a Hermitian subspace in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$L_{W}^{2}(I)\times L_{W}^{2}(I)$\end{document}. It has been proved in 26 that the adjoint of preminimal subspace is the maximal subspace and thus \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$H_{00}^{\ast } = H_{0}^{\ast } = \tilde{H}$\end{document}.…”

“…The definiteness condition A together with the fact that I 2 − A ( t ) is invertible now guarantees the existence of a unique solution for (2.4). If \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$z \in {\mathbb {C}}\backslash {\mathbb {R}}$\end{document}, G. Ren and Y. Shi 26 have shown that the dimension of the defect space of H 0 and also of H 00 are equal to the number of linearly independent square summable solutions of (1.1) or (2.4). Assume that the defect index of H 0 are equal, that is, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\textrm {def}\, H_{0} = (n,n)$\end{document}, where 2 ⩽ n ⩽ 4, then H 0 has selfadjoint extension subspace in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$L_{W}^{2}(I)\times L_{W}^{2}(I)$\end{document} denoted by H if there exists matrices \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\alpha _{1}, \alpha _{2} \in {\mathbb {C}}^{2\times 2}$\end{document} such that with Φ a = (( y 1 , y 2 )( a − 1)) 2 × 2 and Φ ∞ defined in a similar way though with the boundary condition fixed at limiting point, that is, as t → ∞.…”

“…It has been shown in 26 that \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\textrm {def}\, H_{0}$\end{document} is independent of the half‐planes if z is nonreal. In that case, H 0 has a selfadjoint extension subspace H defined by (2.7).…”

“…In that case, H 0 has a selfadjoint extension subspace H defined by (2.7). Moreover, if a closed Hermitian subspace has equal finite defect indices, then all its selfadjoint extension subspaces have the same essential spectrum 26. For point spectrum of these subspaces, every isolated point of the spectrum of selfadjoint subspace is an eigenvalue of the subspace and therefore constitutes the point spectrum.…”

“…Another difficulty that is always encountered in the difference systems is that the minimal operator generated by (1.1) may be neither densely defined nor single valued, its maximal operator may not be well defined and thus the selfadjoint extension operator for minimal operator cannot be discussed by application of von Neumann theory for densely defined Hermitian operators. This, therefore, requires the theory of Hermitian subspaces where the von Neumann theory has been extended in order to discuss the selfadjoint extension of the minimal Hermitian subspaces and for more details see 25–27, 29 and the references cited therein.…”