Thin structure of eigenvalue clusters for non-Hermitian Toeplitz matrices
Let \(\mathbb{T}\) be the unit circle in the complex plane and \(f\in L^\infty(\mathbb{T})\) a complex-valued function on \(\mathbb{T}\) with Fourier coefficients \(f_k\). The authors call a domain \(\Omega\subset\mathbb{C}\) fair if \(\overline\Omega\) is a compact set with connected complement. Assume that the essential range of \(f\) is contained in the union \(\Omega=\bigcup_{k=1}^m \Omega_k\) of fair domains \(\Omega_k\) with disjoint closures. Consider the Toeplitz matrices \(A_n=[f_{i-j}]_ {i,j=1}^n\), and denote by \(\gamma(A_n,\Omega_k,\epsilon)\) the number of eigenvalues of \(A_n\) not lying in the \(\epsilon\)-neighbourhood of \(\Omega_k\). The main result of the paper is that for all \(\epsilon>0\) sufficiently small, \(\gamma(A_n,\Omega_k,\epsilon)/n \to 1-c_k\) as \(n\to\infty\), where \(c_k\) is the normalized arc-length measure of the set \(\{z\in\mathbb{T}: f(z)\in\Omega_k\}\). A similar result involving fair domains \(\Omega_1,\Omega_2\) whose closures have a single common point is also discussed, and a Szegő-type formula for the matrices \(A_n\) is conjectured.
scientific article; zbMATH DE number 1371369