Egor A. Maximenko

Eigenvalues of Hermitian Toeplitz Matrices Generated by Simple-loop Symbols with Relaxed Smoothness

In a sequence of previous works with Albrecht Bottcher, we established higher-order uniform individual asymptotic formulas for the eigenvalues and eigenvectors of large Hermitian Toeplitz matrices generated by symbols satisfying the so-called simple-loop condition, which means that the symbol has only two intervals of monotonicity, its first derivative does not vanish on these intervals, and the second derivative is different from zero at the minimum and maximum points. Moreover, in previous works it was supposed that the symbol belongs to the weighted Wiener algebra W α for α ≥ 4, or satisfies even stronger smoothness conditions. We now use a different technique, which allows us to extend previous results to the case α ≥ 1 with additional smoothness at the minimum and maximum points.

From convergence in distribution to uniform convergence

We present conditions that allow us to pass from the convergence of probability measures in distribution to the uniform convergence of the associated quantile functions. Under these conditions, one can in particular pass from the asymptotic distribution of collections of real numbers, such as the eigenvalues of a family of n-by-n matrices as n goes to infinity, to their uniform approximation by the values of the quantile function at equidistant points. For Hermitian Toeplitz-like matrices, convergence in distribution is ensured by theorems of the Szegő type. Our results transfer these convergence theorems into uniform convergence statements.

Eigenvalues of even very nice Toeplitz matrices can be unexpectedly erratic

It was shown in a series of recent publications that the eigenvalues of \(n\;\times\;n\) 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 Ekstrom, Garoni, and Serra-Capizzano and reveals that the simple-loop condition is essential for the existence of the regular asymptotic expansion.

Radial Toeplitz Operators on the Fock Space and Square-Root-Slowly Oscillating Sequences

In this paper we show that the C*-algebra generated by radial Toeplitz operators with

Vertical Toeplitz Operators on the Upper Half-Plane and Very Slowly Oscillating Functions

L_{\infty }

Vertical symbols, Toeplitz operators on weighted Bergman spaces over the upper half-plane and very slowly oscillating functions

L∞-symbols acting on the Fock space is isometrically isomorphic to the C*-algebra of bounded sequences uniformly continuous with respect to the square-root-metric

C *-Algebra Generated by Angular Toeplitz Operators on the Weighted Bergman Spaces Over the Upper Half-Plane

\rho (j,k)=|\sqrt{j}-\sqrt{k}\,|