@inbook{aaf413212f444720b580198bbb5d1bf7,
title = "Eigenvalues of even very nice Toeplitz matrices can be unexpectedly erratic",
abstract = "It was shown in a series of recent publications that the eigenvalues of n × n Toeplitz matrices generated by so-called simple-loop symbols admit certain regular asymptotic expansions into negative powers of n + 1. On the other hand, recently two of the authors considered the pentadiagonal Toeplitz matrices generated by the symbol g(x) = (2 sin(x/2))4, which does not satisfy the simple-loop conditions, and derived asymptotic expansions of a more complicated form. Here we use these results to show that the eigenvalues of the pentadiagonal Toeplitz matrices do not admit the expected regular asymptotic expansion. This also delivers a counter-example to a conjecture by Ekstr{\"o}m, Garoni, and Serra-Capizzano and reveals that the simple-loop condition is essential for the existence of the regular asymptotic expansion.",
keywords = "Eigenvalue, Spectral asymptotics, Toeplitz matrix",
author = "Mauricio Barrera and Albrecht B{\"o}ttcher and Grudsky, {Sergei M.} and Maximenko, {Egor A.}",
note = "Publisher Copyright: {\textcopyright} 2018, Springer International Publishing AG, part of Springer Nature.",
year = "2018",
doi = "10.1007/978-3-319-75996-8_2",
language = "Ingl{\'e}s",
series = "Operator Theory: Advances and Applications",
publisher = "Springer International Publishing",
pages = "51--77",
booktitle = "Operator Theory",
}