Uniform individual asymptotics for the eigenvalues and eigenvectors of large Toeplitz matrices

The asymptotic behavior of the spectrum of large Toeplitz matrices has been studied for almost one century now. Among this huge work, we can nd the Szeg\H{o} theorems on the eigenvalue distribution and the asymptotics for the determinants, as well as other theorems about the individual asymptotics for the smallest and largest eigenvalues. Results about uniform individual asymptotics for all the eigenvalues and eigenvectors appeared only ve years ago. The goal of the present lecture is to review this area, to talk about the obtained results. This review is based on joint works with Manuel Bogoya, Albrecht B\ ottcher, and Egor Maximenko

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. We here 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.Comment: 28 pages, 7 figure

Toeplitz operators with radial symbols on weighted holomorphic Orlicz space

Peer reviewe

Dynamics of properties of Toeplitz operators on the upper half-plane: Parabolic case

Abstract. We consider Toeplitz operators T (λ) a acting on the weighted Bergman spaces A2 λ (Π), λ ∈ [0, ∞), over the upper half-plane Π, whose symbols depend on θ = arg z. Motivated by the Berezin quantization procedure we study the dependence of the properties of such operators on the parameter of the weight λ and, in particular, under the limit λ → ∞. 1

Fast non-Hermitian Toeplitz eigenvalue computations, joining matrix-less algorithms and FDE approximation matrices

The present work is devoted to the eigenvalue asymptotic expansion of the Toeplitz matrix $T_{n}(a)$ whose generating function $a$ is complex valued and has a power singularity at one point. As a consequence, $T_{n}(a)$ is non-Hermitian and we know that the eigenvalue computation is a non-trivial task in the non-Hermitian setting for large sizes. We follow the work of Bogoya, B\"ottcher, Grudsky, and Maximenko and deduce a complete asymptotic expansion for the eigenvalues. After that, we apply matrix-less algorithms, in the spirit of the work by Ekstr\"om, Furci, Garoni, Serra-Capizzano et al, for computing those eigenvalues. Since the inner and extreme eigenvalues have different asymptotic behaviors, we worked on them independently, and combined the results to produce a high precision global numerical and matrix-less algorithm. The numerical results are very precise and the computational cost of the proposed algorithms is independent of the size of the considered matrices for each eigenvalue, which implies a linear cost when all the spectrum is computed. From the viewpoint of real world applications, we emphasize that the matrix class under consideration includes the matrices stemming from the numerical approximation of fractional diffusion equations. In the final conclusion section a concise discussion on the matter and few open problems are presented.Comment: 21 page