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 SIGMA, 2016, Volume 12, 066, 19 pages (Mi sigma1148)

Periodic GMP Matrices

Benjamin Eichinger

Institute for Analysis, Johannes Kepler University, Linz, Austria

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.

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

 Funding Agency Grant Number Austrian Science Fund P25591-N25 The author was supported by the Austrian Science Fund FWF, project no: P25591-N25.

DOI: https://doi.org/10.3842/SIGMA.2016.066

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Full text: http://www.emis.de/journals/SIGMA/2016/066/
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ArXiv: 1601.07303
MSC: 30E05; 30F15; 47B36; 42C05; 58J53
Received: January 28, 2016; in final form June 29, 2016; Published online July 7, 2016
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Citation: Benjamin Eichinger, “Periodic GMP Matrices”, SIGMA, 12 (2016), 066, 19 pp.

Citation in format AMSBIB
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