Abstract
We introduce and investigate a class of complex semi-infinite banded Toeplitz matrices satisfying the condition that the spectra of their principal submatrices accumulate onto a real interval when the size of the submatrix grows to ∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\infty $$\\end{document}. We prove that a banded Toeplitz matrix belongs to this class if and only if its symbol has real values on a Jordan curve located in C\\{0}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {C}}}{\\setminus }\\{0\\}$$\\end{document}. Surprisingly, it turns out that, if such a Jordan curve is present, the spectra of all the principal submatrices have to be real. The latter claim is also proved for matrices given by a more general symbol. The special role of the Jordan curve is further demonstrated by a new formula for the limiting density of the asymptotic eigenvalue distribution for banded Toeplitz matrices from the studied class. Certain connections between the problem under investigation, Jacobi operators, and the Hamburger moment problem are also discussed. The main results are illustrated by several concrete examples; some of them allow an explicit analytic treatment, while some are only treated numerically.
Highlights
With a Laurent series ∞ a(z) = ak zk (1) k=−∞with complex coefficients ak, one can associate a semi-infinite Toeplitz matrix T (a) whose elements are given by (T (a))i, j = ai− j, ∀i, j ∈ N0.In the theory of Toeplitz matrices, a is referred to as the symbol of T (a) and the set of symbols under consideration may be further restricted depending on what properties of T (a) are studied
For the spectral analysis of T (a), a special role is played by the Wiener algebra which consists of symbols a defined on the unit circle T whose Laurent series (1) is absolutely convergent for z ∈ T
An intimately related subject concerns the asymptotic eigenvalue distribution of Toeplitz matrices for which the most complete results were obtained if the symbol of the matrix is a Laurent polynomial [5]
Summary
An intimately related subject concerns the asymptotic eigenvalue distribution of Toeplitz matrices for which the most complete results were obtained if the symbol of the matrix is a Laurent polynomial [5]. There are almost no criteria for famous non-self-adjoint families (such as Toeplitz, Jacobi, Hankel, Schrödinger, etc.) guaranteeing the reality of their spectra From this point of view and to the best of our knowledge, the present article provides the first relevant results of such a flavor for the class of banded Toeplitz matrices. The last part of the paper contains various numerical illustrations and plots of the densities of the limiting measures and the distributions of eigenvalues in the situations whose complexity does not allow us to treat them explicitly
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