A matrix generalization of a theorem of Szegö (Q2367059)

Let \(\lambda\) be the normalized Lebesgue measure on the unit circle \(\mathbb{T}\) of the complex plane. For a non-negative finite Borel measure \(\mu\) on \(\mathbb{T}\), let \(\mu'\) denote the Radon-Nikodým derivative of the absolutely continuous part of \(\mu\) with respect to \(\lambda\). A well- known theorem of Szegö, which has important applications in prediction theory, states that for \(p\in (0,\infty)\) \[ \exp \int_{\mathbb{T}} \log\mu' d\lambda=\inf \int_{\mathbb{T}} | 1-t|^ p d\lambda \] with the convention \(\exp\int_{\mathbb{T}} \log\mu' d\lambda= 0\) if \(\log\mu'\) is not integrable. Here \(t\) runs through the set of polynomials such that \(t(0)= 0\). \textit{V. N. Zasuhin} [C. R. (Dokl.) Akad. Sci. URSS, N. Sér. 33, 435-437 (1941)] announced and \textit{H. Helson} and \textit{D. Lowdenslager} [Acta Math. 99, 165-202 (1958; Zbl 0082.282)] proved a matrix version of Szegö's theorem in the case \(p=2\). In our paper we extend the result of these authors to arbitrary \(p\in (0,\infty)\).