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Perturbations of orthogonal polynomials with periodic recursion coefficients. (English) Zbl 1194.47031
The authors develop the spectral theory of asymptotically periodic Jacobi and CMV matrices with the focus set on the extension of some basic results concerning asymptotically constant matrices of the above classes. The main difficulty for the asymptotically periodic setting is that now, instead of a single unperturbed operator, one has a manifold (the isospectral thorus) of unperturbed operators. The spectral measures that are close to those of the isospectral thorus correspond to coefficients that approach the isospectral thorus, in general without converging to any particular point therein.
One of the major new results of the paper is the following Theorem 1.2. Let $$J_0$$ be a two-sided periodic Jacobi matrix and $$J$$ a one-sided Jacobi matrix with Jacobi parameters $$\{a_m,b_m\}_{m=1}^\infty$$. If $$\sigma_{ess}(J)=\sigma(J_0)$$ and $$\sigma_{ac}(J)=\sigma(J_0)$$, then $$d_m((a,b), T_{J_0})\to 0$$. Here, $$T_{J_0}$$ is the isospectral torus, and $$d$$ is a suitable distance between pairs of sequences.

##### MSC:
 47B36 Jacobi (tridiagonal) operators (matrices) and generalizations 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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