Determinant calculations for the Block Szego-Widom Limit Theorem

摘要

The Szego-Widom Limit Theorem says that for certain N × N matrix functions φ defined on the unit circle, the asymptotic behavior of the determinants of the block Toeplitz matrices T{sub}n(φ) is given by lim((detT{sub}n(φ)/G[φ]{sup}n)=E[φ], where G[φ] is the geometric mean of detφ and E[φ] is equal to the operator determinant det T(φ)T(φ{sup}(-1)). In the scalar case (N = 1) a more explicit expression for E[φ] exists, while in the matrix case (N > 1) not much is known. In the present paper we are going to establish an explicit expression for E[φ] for 2 × 2 matrix functions φ=αI +βQ where α andβ are scalar functions and Q is a rational matrix function for which Q{sup}2 = 0. It turns out that in comparison with the scalar case, new terms in the expression for E[φ] appear.