The spectrum of a periodic complex Jacobi matrix revisited
type: Item
instance of: scholarly article
title: The spectrum of a periodic complex Jacobi matrix revisited
zbMATH Open document ID: 0966.15004
author: A. Almendral Vázquez
published in: Journal of Approximation Theory
publication date: 2001-01-29T00:00:00Z
review text: 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 \(Q_k(G)\).
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 15A60
zbMATH DE Number: 1523570
zbMATH Keywords: spectrum
zbMATH Keywords: periodic complex Jacobi matrix
zbMATH Keywords: bounded linear operator
MaRDI profile type: Publication
full work available at URL: jath.2000.3480
OpenAlex ID: W1990146597
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DOI: 10.1006/JATH.2000.3480
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description: scientific article; zbMATH DE number 1523570