Determinant computations for some classes of Toeplitz-Hankel matrices

TL;DR

This paper investigates the asymptotic behavior of determinants of specific Toeplitz-Hankel matrices, extending classical results and deriving exact identities relevant to random matrix theory.

Contribution

It introduces new asymptotic formulas and exact determinant identities for classes of Toeplitz-Hankel matrices, generalizing the Strong Szeg"o Limit Theorem.

Findings

Asymptotic expansions similar to the Strong Szeg"o Limit Theorem for Toeplitz-Hankel matrices.

Exact determinant identities analogous to known results for finite Toeplitz matrices.

Applications to statistical quantities in random matrix theory.

Abstract

The purpose of this paper is to compute the asymptotics of determinants of finite sections of operators that are trace class perturbations of Toeplitz operators. For example, we consider the asymptotics in the case where the matrices are of the form $ (a_{i-j} \pm a_{i+j+1-k})_{i,j=0... N-1} $ with $k$ is fixed. We will show that this example as well as some general classes of operators have expansions that are similar to those that appear in the Strong Szeg\"{o} Limit Theorem. We also obtain exact identitities for some of the determinants that are analogous to the one derived independently by Geronimo and Case and by Borodin and Okounkov for finite Toeplitz matrices. These problems were motivated by considering certain statistical quantities that appear in random matrix theory.