2017/02/02 by Boris Shapiro, Shapiro, Boris, František Štampach +1
Computer Science · Mathematics · #15B05 #33C47 #47B36 #Advanced Algebra and Geometry #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical functions and polynomials #Matrix Theory and Algorithms #Point processes and geometric inequalities #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.1702.00741
openalex publication_date 2017/02/02 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
We introduce and investigate a class of complex semi-infinite banded Toeplitz\nmatrices satisfying the condition that the spectra of their principal\nsubmatrices accumulate onto a real interval when the size of the submatrix\ngrows to \∞. We prove that a banded Toeplitz matrix belongs to this class\nif and only if its symbol has real values on a Jordan curve located in\n\ℂ\∖ 0 . Surprisingly, it turns out that, if such a Jordan\ncurve is present, the spectra of all the submatrices have to be real. The\nlatter claim is also proven for matrices given by a more general symbol.\nFurther, the limiting eigenvalue distribution of a real banded Toeplitz matrix\nis related to the solution of a determinate Hamburger moment problem. We use\nthis to derive a formula for the limiting measure using a parametrization of\nthe Jordan curve. We also describe a Jacobi operator, whose spectral measure\ncoincides with the limiting measure. We show that this Jacobi operator is a\ncompact perturbation of a tridiagonal Toeplitz matrix. Our main results are\nillustrated by several concrete examples; some of them allow an explicit\nanalytic treatment, while some are only treated numerically.\n Update: The proof of Theorem 8 contains an error. An erratum is attached in\nthe end\n