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

SIGMA 12 (2016), 066, 19 pages      arXiv:1601.07303
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

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.

pdf (437 kb)   tex (25 kb)


  1. Ahiezer N.I., Orthogonal polynomials on several intervals, Soviet Math. Dokl. 1 (1960), 989-992.
  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.
  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.
  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, math.CA/0204300.
  5. Damanik D., Killip R., Simon B., Perturbations of orthogonal polynomials with periodic recursion coefficients, Ann. of Math. 171 (2010), 1931-2010, math.SP/0702388.
  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.
  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, 4th ed., Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 2013.
  9. Hasumi M., Hardy classes on infinitely connected Riemann surfaces, Lecture Notes in Math., Vol. 1027, Springer-Verlag, Berlin, 1983.
  10. Heins M., Hardy classes on Riemann surfaces, Lecture Notes in Math., Vol. 98, Springer-Verlag, Berlin - New York, 1969.
  11. Hendriksen E., Nijhuis C., Laurent-Jacobi matrices and the strong Hamburger moment problem, Acta Appl. Math. 61 (2000), 119-132.
  12. Katsnelson V., On a family of Laurent polynomials generated by $2\times2$ matrices, Complex Anal. Oper. Theory, to appear, arXiv:1507.06101.
  13. Killip R., Simon B., Sum rules for Jacobi matrices and their applications to spectral theory, Ann. of Math. 158 (2003), 253-321, math-ph/0112008.
  14. Krichever I.M., Algebraic curves and nonlinear difference equations, Russ. Math. Surv. 33 (1978), no. 4, 255-256.
  15. Levin B.Ja., Distribution of zeros of entire functions, Translations of Mathematical Monographs, Vol. 5, Amer. Math. Soc., Providence, R.I., 1980.
  16. Pommerenke C., Über die analytische Kapazität, Arch. Math. (Basel) 11 (1960), 270-277.
  17. Pommerenke C., On the Green's function of Fuchsian groups, Ann. Acad. Sci. Fenn. Ser. A I Math. 2 (1976), 409-427.
  18. Simon B., CMV matrices: five years after, J. Comput. Appl. Math. 208 (2007), 120-154, math.SP/0603093.
  19. Simon B., Szegő's theorem and its descendants. Spectral theory for $L^2$ perturbations of orthogonal polynomials, M.B. Porter Lectures, Princeton University Press, Princeton, NJ, 2011.
  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.
  21. Teschl G., Jacobi operators and completely integrable nonlinear lattices, Mathematical Surveys and Monographs, Vol. 72, Amer. Math. Soc., Providence, RI, 2000.
  22. Verblunsky S., On positive harmonic functions: a contribution to the algebra of Fourier series, Proc. London Math. Soc. S2-38 (1935), 125-157.
  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.
  24. Widom H., ${\cal H}_{p}$ sections of vector bundles over Riemann surfaces, Ann. of Math. 94 (1971), 304-324.
  25. Yuditskii P., Killip-Simon problem and Jacobi flow on GMP matrices, arXiv:1505.00972.

Previous article  Next article   Contents of Volume 12 (2016)