Essential spectrum of singular discrete linear Hamiltonian systems (Q2792259)

The authors characterize all elements of the essential spectrum of the minimal linear relation associated with the linear Hamiltonian difference system NEWLINE\[NEWLINE\mathcal J\Delta y_k-P_k R(y_k)=\lambda\,W_k R(y_k) \tag{1}NEWLINE\]NEWLINE by using square summable solutions of System (1) or of the corresponding nonhomogeneous system. In (1) the index \(k\) belongs to a finite or infinite discrete interval, \(\lambda\in\mathbb{C}\), \(\mathcal{J}:=\begin{pmatrix} 0 & -I\\ I & 0\end{pmatrix}\) with \(I\) being the \(n\times n\) identity matrix, \(\Delta\) denotes the forward difference operator, \(P_k\) are \(2n\times 2n\) Hermitian matrices for all \(k\), \(R(\cdot)\) is the partial shift operator, and \(W_k\) are \(2n\times 2n\) positive semidefinite weight matrices in block diagonal form for all \(k\). Consequently, they obtain analogous results also for any \(2m\)-order vector-valued Sturm-Liouville difference equation.