On a Toeplitz Determinant Identity of Borodin and Okounkov

The authors of the title proved in [2] an elegant identity expressing a Toeplitz determinant in terms of the Fredholm determinant of an infinite matrix which (although not described as such) is the product of two Hankel matrices. The proof used combinatorial theory, in particular a theorem of Gessel expressing a Toeplitz determinant as a sum over partitions of products of Schur functions. The purpose of this note is to give two other proofs of the identity. The first uses an identity of the second author [4] for the quotient of Toeplitz determinants in which the same product of Hankel matrices appears and the second, which is more direct and extends the identity to the case of block Toeplitz determinants, consists of carrying the first author’s collaborative proof [1] of the strong Szegö limit theorem one step further. We begin with the statement of the identity of [2], changing notation slightly. If φ is a function on the unit circle with Fourier coefficients φk then Tn(φ) denotes the Toeplitz matrix (φi−j)i,j=0,···,n−1 and Dn(φ) its determinant. Under general conditions φ has a representation φ = φ+ φ− where φ+ (resp. φ−) extends to a nonzero analytic function in the interior (resp. exterior) of the circle. We assume that φ has geometric mean 1, and normalize φ± so that φ+(0) = φ−(∞) = 1. Define the infinite matrices Un and Vn acting on l (Z), where Z = {0, 1, · · ·}, by