Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 066, 19 pages      arXiv:1601.07303      https://doi.org/10.3842/SIGMA.2016.066
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications

Periodic GMP Matrices

Benjamin Eichinger
Institute for Analysis, Johannes Kepler University, Linz, Austria

Received January 28, 2016, in final form June 29, 2016; Published online July 07, 2016

Abstract
We recall criteria on the spectrum of Jacobi matrices such that the corresponding isospectral torus consists of periodic operators. Motivated by those known results for Jacobi matrices, we define a new class of operators called GMP matrices. They form a certain Generalization of matrices related to the strong Moment Problem. This class allows us to give a parametrization of almost periodic finite gap Jacobi matrices by periodic GMP matrices. Moreover, due to their structural similarity we can carry over numerous results from the direct and inverse spectral theory of periodic Jacobi matrices to the class of periodic GMP matrices. In particular, we prove an analogue of the remarkable ''magic formula'' for this new class.

Key words: spectral theory; periodic Jacobi matrices; bases of rational functions; functional models.

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