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On the convergence of bounded \(J\)-fractions on the resolvent set of the corresponding second order difference operator. (English) Zbl 0931.30002

A \(J\)-fraction is a continued fraction associated with the second order difference equation \(zy_n(z)=a_{n-1}y_{n-1}(z)+b_ny_n(z)+a_ny_{n+1}(z)\), \(n\geq 0\). Considering \(y=(y_n)\) as a sequence in \(\ell^2\), this equation can be written as \(zy=Ay\). With respect to the natural basis \((e_k)_{k=0}^\infty\) in \(\ell^2\), \(A\) has a symmetric tridiagonal Jacobi matrix representation. The Weyl function of \(A\) is then given by the inner product in \(\ell^2\): \(\phi(z)=\langle (zI-A)^{-1}e_0,e_0\rangle\). The convergence of the \(J\)-fraction are Padé approximants for \(\phi\). They converge uniformly to \(\phi\) in the neighborhood of infinity. In this paper, convergence in capacity and the rate of convergence is established in the unbounded connected component \(\Omega_0\) of the resolvent set \(\Omega\) of \(A\). The proof relies on a generalization of Widom’s result concerning the number of poles of \(\phi\) in \(\Omega_0\), and on the ratio and root asymptotics for the Padé denominators for bounded \(J\)-fractions. Relations with Green’s function for the resolvent set of \(A\) are given. However, even for a bounded operator \(A\), one may have \(\Omega_0\neq\Omega\). An investigation of isolated points in the spectrum of \(A\), their relation to the poles and isolated singularities of the Weyl function and so called spurious poles of the Padé approximants, leads to the proof of the Baker-Gammel-Wills conjecture which says that in the case of a countable spectrum one has local uniform convergence of a subsequence of Padé approximants in the maximal domain of analyticity of the Weyl function.

MSC:

30B70 Continued fractions; complex-analytic aspects
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