Geodesic
Periodic GMP Matrices
Symmetry, integrability and geometry: methods and applications, Tome 12 (2016) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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. 
@article{SIGMA_2016_12_a65,
     author = {Benjamin Eichinger},
     title = {Periodic {GMP} {Matrices}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2016},
     volume = {12},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a65/}
}
TY  - JOUR
AU  - Benjamin Eichinger
TI  - Periodic GMP Matrices
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2016
VL  - 12
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a65/
LA  - en
ID  - SIGMA_2016_12_a65
ER  - 
%0 Journal Article
%A Benjamin Eichinger
%T Periodic GMP Matrices
%J Symmetry, integrability and geometry: methods and applications
%D 2016
%V 12
%U http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a65/
%G en
%F SIGMA_2016_12_a65
Benjamin Eichinger. Periodic GMP Matrices. Symmetry, integrability and geometry: methods and applications, Tome 12 (2016). http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a65/

[1] Ahiezer N. I., “Orthogonal polynomials on several intervals”, Soviet Math. Dokl., 1 (1960), 989–992 | MR

[2] Aptekarev A. I., “Asymptotic properties of polynomials orthogonal on a system of contours, and periodic motions of Toda chains”, Math. USSR Sb., 125(167) (1984), 233–260 | DOI | MR

[3] Baratchart L., Kupin S., Lunot V., Olivi M., “Multipoint Schur algorithm and orthogonal rational functions, I: Convergence properties”, J. Anal. Math., 114 (2011), 207–253, arXiv: 0812.2050 | DOI | MR | Zbl

[4] Cantero M. J., Moral L., Velázquez L., “Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle”, Linear Algebra Appl., 362 (2003), 29–56, arXiv: math.CA/0204300 | DOI | MR | Zbl

[5] Damanik D., Killip R., Simon B., “Perturbations of orthogonal polynomials with periodic recursion coefficients”, Ann. of Math., 171 (2010), 1931–2010, arXiv: math.SP/0702388 | DOI | MR | Zbl

[6] Eichinger B., Puchhammer F., Yuditskii P., “Jacobi flow on SMP matrices and Killip–Simon problem on two disjoint intervals”, Comput. Methods Funct. Theory, 16 (2016), 3–41 | DOI | MR | Zbl

[7] Eichinger B., Yuditskii P., Killip–Simon problem and Jacobi flow on GSMP matrices, arXiv: 1412.1702

[8] Golub G. H., Van Loan C. F., Matrix computations, Johns Hopkins Studies in the Mathematical Sciences, 4th ed., Johns Hopkins University Press, Baltimore, MD, 2013 | MR | Zbl

[9] Hasumi M., Hardy classes on infinitely connected Riemann surfaces, Lecture Notes in Math., 1027, Springer-Verlag, Berlin, 1983 | DOI | MR | Zbl

[10] Heins M., Hardy classes on Riemann surfaces, Lecture Notes in Math., 98, Springer-Verlag, Berlin–New York, 1969 | DOI | MR | Zbl

[11] Hendriksen E., Nijhuis C., “Laurent–Jacobi matrices and the strong Hamburger moment problem”, Acta Appl. Math., 61 (2000), 119–132 | DOI | MR | Zbl

[12] Katsnelson V., “On a family of Laurent polynomials generated by $2\times2$ matrices”, Complex Anal. Oper. Theory (to appear) , arXiv: 1507.06101 | DOI | MR

[13] Killip R., Simon B., “Sum rules for Jacobi matrices and their applications to spectral theory”, Ann. of Math., 158 (2003), 253–321, arXiv: math-ph/0112008 | DOI | MR | Zbl

[14] Krichever I. M., “Algebraic curves and nonlinear difference equations”, Russ. Math. Surv., 33:4 (1978), 255–256 | DOI | MR | Zbl

[15] Levin B. Ja., Distribution of zeros of entire functions, Translations of Mathematical Monographs, 5, Amer. Math. Soc., Providence, R.I., 1980 | MR

[16] Pommerenke C., “Über die analytische Kapazität”, Arch. Math. (Basel), 11 (1960), 270–277 | DOI | MR | Zbl

[17] Pommerenke C., “On the Green's function of Fuchsian groups”, Ann. Acad. Sci. Fenn. Ser. A I Math., 2 (1976), 409–427 | DOI | MR | Zbl

[18] Simon B., “CMV matrices: five years after”, J. Comput. Appl. Math., 208 (2007), 120–154, arXiv: math.SP/0603093 | DOI | MR | Zbl

[19] Simon B., Szeg{ő}'s theorem and its descendants. Spectral theory for $L^2$ perturbations of orthogonal polynomials, M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 2011 | MR

[20] Sodin M. L., Yuditskii P. M., “Infinite-dimensional real problem of Jacobi inversion and Hardy spaces of character-automorphic functions”, Dokl. Akad. Nauk, 335 (1994), 161–163 | MR | Zbl

[21] Teschl G., Jacobi operators and completely integrable nonlinear lattices, Mathematical Surveys and Monographs, 72, Amer. Math. Soc., Providence, RI, 2000 | MR | Zbl

[22] Verblunsky S., “On positive harmonic functions: a contribution to the algebra of Fourier series”, Proc. London Math. Soc., S2-38 (1935), 125–157 | DOI | MR

[23] Volberg A., Yuditskii P., “Kotani–Last problem and Hardy spaces on surfaces of Widom type”, Invent. Math., 197 (2014), 683–740, arXiv: 1210.7069 | DOI | MR | Zbl

[24] Widom H., “${\cal H}_{p}$ sections of vector bundles over Riemann surfaces”, Ann. of Math., 94 (1971), 304–324 | DOI | MR | Zbl

[25] Yuditskii P., Killip–Simon problem and Jacobi flow on GMP matrices, arXiv: 1505.00972