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Toeplitz matrices and determinants with Fisher-Hartwig symbols. (English) Zbl 0592.47016
The authors obtain an extension of the classical Szegö limit theorem for the asymptotic behavior of Toeplitz determinants \(D_ n(a)=\det (a_{j-k})^{\infty}_{j,k=0}\) as \(n\to \infty\). In particular, they consider ”singular” symbols of the form \(a(t)=\prod^{R}_{r=1}| t- t_ r|^{2\alpha_ r}(-t)_{t_ r}^{\beta_ r}b(t)\) where \(t_ 1,...,t_ R\) are distinct points on the unit circle \({\mathbb{T}}\), the function b(t) is non-zero and sufficiently smooth on \({\mathbb{T}}\) with ind(b)\(\neq 0\), \(| Re \alpha_ r| <1/2\), \(| Re \beta_ r| <1/2\), and \((-t)_{t_ r}^{\beta_ r}\) is defined by \(\exp \{i\beta_ r \arg (-t/t_ r)\}\), with \(| \arg (-t/t_ r)| <\pi\). Such symbols arise in physical applications and were studied by Fisher and Hartwig [Adv. Chem. Phys. 15, 333-353 (1968)], where it was conjectured that the \(D_ n(a)\) are asymptotically equal to \(G(b)^ nn^ q\tilde E(a)\), where \(q=\sum (A^ 2_ r-b^ 2_ r)\) and G(b), \(\tilde E(\)a) are certain non-zero constants. By considering the Toeplitz operator T(A) as acting on weighted Hilbert spaces, the present authos prove this conjecture for the symbols described above.
Reviewer: J.Butz

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
15A15 Determinants, permanents, traces, other special matrix functions
Full Text: DOI
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