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Toeplitz operators and determinants generated by symbols with one Fisher- Hartwig singularity. (English) Zbl 0613.47024
Let T be the unit circle. Given a function a in $$L^ 1(T)$$ denote by $$T_ n(a)$$ the finite Toeplitz matrix $$(a_{j-k})^ n_{j,k=0}$$ $$[a_ k$$ is the k-th Fourier coefficient of a] and by $$D_ n(a)$$ the determinant of $$T_ n(a)$$ $$(n=0,1,2,...)$$. The paper focuses attention on the symbols of the form $(*)\quad a(t)=(1-t_ 0/t)^{\delta}(1- t/t_ 0)^{\gamma}b(t)\quad (| t| =1),$ where $$t_ 0\in T$$, $$\delta$$ and $$\gamma$$ are complex numbers satisfying $$Re \gamma+Re \delta >1$$, and b is a sufficiently smooth function which does not vanish on T and has winding number zero. The problems of interest are the asymptotic behaviour of the determinants $$D_ n(a)$$ as $$n\to \infty$$ and the asymptotic behaviour of the entries of the inverses $$T_ n^{-1}(a)$$ (if they exist).
M. E. Fisher and R. E. Hartwig [Toeplitz determinants: some applications, theorems, and conjectures, Adv. Chem. Phys. 15, 333-353 (1969)] conjectured that for symbols of type (*) $D_ n(a)\sim G(b)^{n+1}n^{\gamma \delta}\tilde E(b,t_ 0,\delta,\gamma)\quad (n\to \infty)$ with certain non-zero constants $$G(b)$$ and $$\tilde E(...)$$. They proved if for $$b(t)\equiv 1$$, $$\gamma =-\delta,| Re \gamma | <$$. The present authors establish the conjecture for the case $$Re \gamma+Re \delta \geq 0$$, $$Re \gamma>-1$$, $$Re \delta>-1$$. The two questions raised above are approached by the finite section method for Toeplitz operators generated by symbols of the form (*) on certain weighted $$\ell^ p$$ spaces.
Reviewer: K.Seddighi

##### MSC:
 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
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