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Finite difference operators with a finite band spectrum. (English) Zbl 1097.47031

In the very beginning of the paper, the authors discuss a certain formal scheme that leads from finite difference ergodic (almost periodic) operators to the related Riemann surfaces and to the functional realization of the given operators in the Hardy spaces on it. Such steps generalize the similar construction for periodic operators of P. van Moerbeke and D. Mumford [Acta Math. 143, 93–154 (1979; Zbl 0502.58032)]. However, these considerations in the present paper have a partially formal and a partially heuristic nature. Starting from the other end (that is, from the Riemann surface and the functional spaces on it), the authors associate to such collections of data an almost periodic multi-diagonal operator. This allows them to construct examples of almost periodic operators with varying spectral properties. The main point is that the authors parametrize the corresponding Riemann surfaces in terms of so-called branching divisors, which make all considerations really constructive. Finally, the authors discuss the uniqueness problem for this sort of functional realizations.
The main theorem describes uniqueness in the following terms: the model is unique if and only if two specified functions separate points of the spectral Riemann surface. The authors provide the following non-uniqueness example: let \(J_0\) be periodic Jacobi matrix. Then there exists a polynomial \(T\) such that \(T(J_0)=S^d+S^{-d}\), where \(d=\deg(T)\) and \(S\) is the shift operator. Due to this remark, one can associate to the multi-diagonal matrix \(S^d+S^{-d}\) with constant coefficients many different functional realizations based on the standard functional realizations of periodic Jacobi matrices.

MSC:

47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
47A45 Canonical models for contractions and nonselfadjoint linear operators
47B39 Linear difference operators

Citations:

Zbl 0502.58032
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