Dario Giandinoto: Spectral asymptotics for two kinds of Toeplitz-like matrices

Abstract: 

The first part of this talk will focus on block Toeplitz matrices with real asymptotic spectrum. I will introduce the Schmidt-Spitzer theorem on the asymptotic spectrum of Toeplitz matrices, and explain how it can be used to obtain a necessary condition for the reality of said spectrum. I will then present how to generalize this result to the case of block Toeplitz matrices, i.e. Toeplitz matrices whose entries are matrices themselves.

In the second part, we will move on to another generalization of Toeplitz matrices, the so-called KMS matrices. Instead of having constant entries on the diagonals like in the Toeplitz case, these matrices have slow-varying entries on their diagonals. I will show some interesting numerical experiments on the spectrum of such matrices and then obtain the spectral asymptotics in a particular case. Finally, I will explain how it could be possible to use this result to obtain the asymptotics for the spectrum of a randomized version of KMS matrices.