The spectrum of a periodic complex Jacobi matrix revisited
Let \(G\) be a complex periodic Jacobi matrix of period \(k\), i.e. \[ G=\begin{pmatrix} b^{(0)}&a^{(1)}&0&\cdots\\ a^{(1)}&b^{(1)}&a^{(2)}&\cdots\\ 0&a^{(2)}&b^{(2)}&\cdots\\ \vdots&\vdots&\vdots&\ddots\end{pmatrix}, \] where \(a^{(n)}=a^{(j)}\), \(b^{(n)}=b^{(j)}\) for \(n\equiv j\pmod k\). This matrix defines a bounded linear operator on \(\ell^2\). In this paper, the author constructs a polynomial \(Q_k(z)\), such that \(Q_k(G)\) is a block tridiagonal matrix of the form \[ Q_k(G)=\begin{pmatrix} O_k&I_k&O_k&\cdots\\ I_k&O_k&I_k&\cdots\\ O_k&I_k&O_k&\cdots\\ \vdots&\vdots&\vdots&\ddots\end{pmatrix}, \] where \(I_k\) and \(O_k\) denote the identity and null matrices of order \(k\) respectively. Thus the study of the spectrum of \(G\) can be reduced to that of \(Q_k(G)\).
scientific article; zbMATH DE number 1523570
scientific article