Determinant computations for some classes of Toeplitz-Hankel matrices

arXiv:0804.3073v1 [math.FA] 18 Apr 2008

Determinant computations for some classes of Toeplitz-Hankel matrices Estelle L. Basor∗ Department of Mathematics California Polytechnic State University San Luis Obispo, CA 93407, USA

Torsten Ehrhardt† Department of Mathematics University of California Santa Cruz, CA-95064, USA

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 (ai−j ± ai+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.

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Introduction

There is a fundamental connection between determinants of certain matrices and random matrix ensembles. For example, one can consider the Haar measure as a probability measure on the set of N × N unitary matrices. This set of matrices along with this measure is usually referred to as the Circular Unitary Ensemble (CUE). From this measure one can then compute the density for the distribution of the eigenvalues eiθ1 , . . . , eiθN of the matrices. The density turns out to be a constant times Y |eiθj − eiθk |2 . j 2. Then T (t−n )M(tm + t−m ) = T (t−n+1 )T (t−1 )M(tm + t−m ) which by the induction arguments equals T (t−n+1 )M(tm−1 + t−m−1 ) = T (t−n+1)M(tm−1 + t−m+1 ) + T (t−n+1 )T (t−m−1 − t−m+1 ) = M(tm−n + t−m−n ) + T (t−m−n − t−n−m+2 ) = M(tm−n + t−n−m+2 ) as desired. Finally, the remarks at the beginning of the section show that K(tn ) = Pn K(tn ) and since the trace norm of Pn is n we have that the trace norm of K(tn ) is bounded by some constant times n. Thus by our choice of Banach algebra, our relations define a bounded linear operator for all function in F ℓ11 . 2 Our final remark is that it is easy to find examples that satisfy the hypothesis of our last theorem. In fact, we can take x in our definition of K to be the vector with a single one in the jth entry and zero otherwise. In fact, our four concrete examples are when the first row of K(t) is either the zero vector, ±e0 or e1 .

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