Abstract
AbstractThe Szegő–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 Tn (φ) is given by equation image where G [φ ] is the geometric mean of det φ and E [φ ] is equal to the operator determinant det T (φ)T (φ–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 and β are scalar functions and Q is a rational matrix function for which Q2 = 0. It turns out that in comparison with the scalar case, new terms in the expression for E [φ ] appear. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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