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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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