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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

##### MSC:
 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators 15A15 Determinants, permanents, traces, other special matrix functions
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##### References:
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