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Advances in Harmonic Analysis and Operator Theory pp 101–122Cite as

Birkhäuser

The Factorization Problem: Some Known Results and Open Questions

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Part of the book series: Operator Theory: Advances and Applications ((OT,volume 229))

Abstract

This is a concise survey of some results and open problems concerning Wiener-Hopf factorization and almost periodic factorization of matrix functions. Several classes of discontinuous matrix functions are considered. Also sketched is the abstract framework which unifies the two types of factorization.

Mathematics Subject Classification (2010). Primary 47A68; Secondary 30E25, 43A75, 45F15, 47B35 .

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References

  1. V. Adukov, Wiener-Hopf operators on a subsemigroup of a discrete torsion free abelian group, Integral Equations and Operator Theory 16 (1993), 305–332.

    Article  MathSciNet  MATH  Google Scholar 

  2. V. Adukov, On invertibility of matrix Wiener-Hopf operator on discrete linearly ordered abelian group, Integral Equations and Operator Theory 23 (1995), 373–386.

    Google Scholar 

  3. S. Avdonin, A. Bulanova, andW. Moran, Construction of sampling and interpolating sequences for multi-band signals. The two-band case, Int. J. Appl. Math. Comput. Sci. 17 (2007), 143–156.

    Google Scholar 

  4. R. Balan and I. Krishtal, An almost periodic noncommutative Wiener’s lemma, J. Math. Anal. Appl. 370 (2010), 339–349.

    Article  MathSciNet  MATH  Google Scholar 

  5. M.A. Bastos, A. Bravo, Yu.I. Karlovich, and I.M. Spitkovsky, Constructive factorization of some almost periodic triangular matrix functions with a quadrinomial off diagonal entry, J. Math. Anal. Appl. 376 (2011), 625–640.

    Article  MathSciNet  MATH  Google Scholar 

  6. M.A. Bastos, Yu.I. Karlovich, A.F. dos Santos, and P.M. Tishin, The corona theorem and the canonical factorization of triangular 𝐴𝑃-matrix functions – Effective criteria and explicit formulas, J. Math. Anal. Appl. 223 (1998), 523–550.

    Article  MathSciNet  MATH  Google Scholar 

  7. M.A. Bastos, Yu.I. Karlovich, and B. Silbermann, Toeplitz operators with symbols generated by slowly oscillating and semi-almost periodic matrix functions, Proc. London Math. Soc. 89 (2004), 697–737.

    Article  MathSciNet  MATH  Google Scholar 

  8. M.A. Bastos, Yu.I. Karlovich, I.M. Spitkovsky, and P.M. Tishin, On a new algorithm for almost periodic factorization, Oper. Theory Adv. Appl. 103 (1998), 53–74.

    MathSciNet  Google Scholar 

  9. S. Bochner and R.S. Phillips, Absolutely convergent Fourier expansion for noncommutative normed rings, Ann. of Math. 43 (1942), 409–418.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Böttcher and Yu.I. Karlovich, Carleson curves, Muckenhoupt weights, and Toeplitz operators, Birkhäuser Verlag, Basel and Boston 1997.

    Google Scholar 

  11. A. Böttcher, Yu.I. Karlovich, and I.M. Spitkovsky, Toeplitz operators with semialmost periodic symbols on spaces with Muckenhoupt weight, Integral Equations and Operator Theory 18 (1994), 261–276.

    Google Scholar 

  12. A. Böttcher, Yu.I. Karlovich, and I.M. Spitkovsky, Convolution operators and factorization of almost periodic matrix functions, Birkhäuser Verlag, Basel and Boston 2002.

    Google Scholar 

  13. A. Böttcher, S. Roch, B. Silbermann, and I. Spitkovsky, A Gohberg-Krupnik-Sarason symbol calculus for algebras of Toeplitz, Hankel, Cauchy, and Carleman operators, Oper. Theory Adv. Appl. 48 (1990), 189–234.

    Google Scholar 

  14. A. Böttcher, B. Silbermann, and I.M. Spitkovskiǐ, Toeplitz operators with piecewise quasisectorial symbols, Bull. London Math. Soc. 22 (1990), 281–286.

    Google Scholar 

  15. A. Böttcher and I. Spitkovsky, Toeplitz operators with 𝑃𝑄𝐶 symbols on weighted Hardy spaces, J. Functional Analysis 97 (1991), 194–214.

    Google Scholar 

  16. A. Brudnyi, L. Rodman, and I.M. Spitkovsky, Factorization versus invertibility of matrix functions on compact abelian groups, Oper. Theory Adv. Appl. 218 (2012), 225–240.

    MathSciNet  Google Scholar 

  17. A. Brudnyi, L. Rodman, and I.M. Spitkovsky, Non-denseness of factorable matrix functions, J. Functional Analysis 261 (2011), 1969–1991.

    Google Scholar 

  18. M.S. Budjanu and I.C. Gohberg, General theorems on the factorization of matrixvalued functions. I. Fundamental theorems, Amer. Math. Soc. Transl. 102 (1973), 1–14.

    Google Scholar 

  19. M.S. Budjanu and I.C. Gohberg, General theorems on the factorization of matrix-valued functions. II. Some tests and their consequences, Amer. Math. Soc. Transl. 102 (1973), 15–26.

    Google Scholar 

  20. M.C. Câmara and C. Diogo, Invertibility of Toeplitz operators and corona conditions in a strip, J. Math. Anal. Appl. 342 (2008), 1297–1317.

    Google Scholar 

  21. M.C. Câmara, C. Diogo, and Yu.I. Karlovich, and I.M. Spitkovsky, Factorizations, Riemann-Hilbert problems and the corona theorem, arXiv:1103.1935v1 [math.FA] (2011), 1–32, accepted by the Journal of the London Mathematical Society.

    Google Scholar 

  22. M.C. Câmara, C. Diogo, and L. Rodman, Fredholmness of Toeplitz operators and corona problems, J. Functional Analysis 259 (2010), 1273–1299.

    Google Scholar 

  23. M.C. Câmara, Yu.I. Karlovich, and I.M. Spitkovsky, Kernels of asymmetric Toeplitz operators and applications to almost periodic factorization, Complex Analysis and Operator Theory, to appear.

    Google Scholar 

  24. M.C. Câmara, Yu.I. Karlovich, and I.M. Spitkovsky, Almost periodic factorization of some triangular matrix functions, Oper. Theory Adv. Appl. 190 (2009), 171–190.

    Google Scholar 

  25. M.C. Câmara, Yu.I. Karlovich, and I.M. Spitkovsky, Constructive almost periodic factorization of some triangular matrix functions, J. Math. Anal. Appl. 367 (2010), 416–433.

    Google Scholar 

  26. M.C. Câmara and M.C. Martins, Explicit almost periodic factorization for a class of triangular matrix functions. J. Anal. Math. 103 (2007), 221–260.

    Google Scholar 

  27. M.C. Câmara, A.F. dos Santos, andM.C. Martins, A new approach to factorization of a class of almost-periodic triangular symbols and related Riemann-Hilbert problems, J. Functional Analysis 235 (2006), 559–592.

    Google Scholar 

  28. M.C. Câmara, A.F. dos Santos, and P.F. dos Santos, Matrix Riemann-Hilbert problems and factorization on Riemann surfaces, J. Functional Analysis 255 (2008), 228–254.

    Google Scholar 

  29. K.F. Clancey and I. Gohberg, Factorization of matrix functions and singular integral operators, Birkhäuser Verlag, Basel and Boston 1981.

    Google Scholar 

  30. C. Corduneanu, Almost periodic functions, J. Wiley & Sons, 1968.

    Google Scholar 

  31. A. Devinatz and M. Shinbrot, General Wiener-Hopf operators, Trans. Amer. Math. Soc. 145 (1969), 467–494.

    Article  MathSciNet  MATH  Google Scholar 

  32. R.G. Douglas, Banach algebra techniques in operator theory, Academic Press, New York 1972.

    Google Scholar 

  33. T. Ehrhardt, C.V.M. van der Mee, L. Rodman, and I.M. Spitkovsky, Factorizations in weighted Wiener algebras on ordered abelian groups, Integral Equations and Operator Theory 58 (2007), 65–86.

    Article  MathSciNet  MATH  Google Scholar 

  34. T. Ehrhardt and F.-O. Speck, Transformation techniques towards the factorization of non-rational 2 × 2 matrix functions, Linear Algebra Appl. 353 (2002), 53–90.

    Article  MathSciNet  MATH  Google Scholar 

  35. T. Ehrhardt and I.M. Spitkovsky, Factorization of piecewise-constant matrix functions, and systems of linear differential equations, St. Petersburg Math. J. 13 (2002), 939–991.

    MathSciNet  MATH  Google Scholar 

  36. I. Feldman, I. Gohberg, and N. Krupnik, An explicit factorization algorithm, Integral Equations and Operator Theory 49 (2004), 149–164.

    Article  MathSciNet  MATH  Google Scholar 

  37. M. Gelfand and I.M. Spitkovsky, Almost periodic factorization: Applicability of the division algorithm, Voronezh Winter Mathematical Schools (P. Kuchment and V. Lin, eds.), vol. 184, Advances in the Mathematical Sciences, no. 2, Amer. Math. Soc., 1998, pp. 97–109.

    Google Scholar 

  38. I. Gohberg, The factorization problem in normed rings, functions of isometric and symmetric operators, and singular integral equations, Uspehi Mat. Nauk 19 (1964), 71–124 (in Russian).

    Google Scholar 

  39. I. Gohberg, M.A. Kaashoek, and I.M. Spitkovsky, An overview of matrix factorization theory and operator applications, Oper. Theory Adv. Appl. 141 (2003), 1–102.

    MathSciNet  Google Scholar 

  40. I. Gohberg and M.G. Krein, Systems of integral equations on a half-line with kernel depending upon the difference of the arguments, Amer.Math. Soc. Transl. 14 (1960), 217–287.

    MathSciNet  Google Scholar 

  41. I. Gohberg and N. Krupnik, One-dimensional linear singular integral equations, Vols. 1 and 2, Birkhäuser Verlag, Basel and Boston 1992.

    Google Scholar 

  42. G. Heinig and B. Silbermann, Factorization of matrix functions in algebras of bounded functions, Oper. Theory Adv. Appl. 14 (1984), 157–177.

    MathSciNet  Google Scholar 

  43. G. Janashia, E. Lagvilava, and L. Ephremidze, A new method of matrix spectral factorization, IEEE Trans. Inform. Theory 57 (2011), 2318–2326.

    Article  MathSciNet  Google Scholar 

  44. A.Yu. Karlovich, Fredholmness of singular integral operators with piecewise continuous coefficients on weighted Banach function spaces, J. Integral Equations Appl. 15 (2003), 263–320.

    Article  MathSciNet  MATH  Google Scholar 

  45. A.Yu. Karlovich, Singular integral operators on variable Lebesgue spaces over arbitrary Carleson curves, Oper. Theory Adv. Appl. 202 (2010), 321–336.

    Google Scholar 

  46. A.Yu. Karlovich and I.M. Spitkovsky, On singular integral operators with semi-almost periodic coefficients on variable Lebesgue spaces, J. Math. Anal. Appl. 384 (2011), 706–725.

    Google Scholar 

  47. Yu.I. Karlovich, Approximation approach to canonical 𝐴𝑃𝑊 factorability, Izv. Vuzov., Sev.-Kavk. Region, 2005, pp. 143–151 (in Russian).

    Google Scholar 

  48. Yu.I. Karlovich and I.M. Spitkovsky, On the Noether property for certain singular integral operators with matrix coefficients of class 𝑆𝐴𝑃 and the systems of convolution equations on a finite interval connected with them, SovietMath. Dokl. 27 (1983), 358–363.

    Google Scholar 

  49. Yu.I. Karlovich and I.M. Spitkovsky, Factorization of almost periodic matrix functions and Fredholm theory of Toeplitz operators with semi almost periodic matrix symbols, Linear and Complex Analysis Problem Book: 199 research problems, Lecture Notes in Mathematics 1043 (1984), 279–282.

    Google Scholar 

  50. Yu.I. Karlovich and I.M. Spitkovsky, Factorization of almost periodic matrix-valued functions and the Noether theory for certain classes of equations of convolution type, Mathematics of the USSR, Izvestiya 34 (1990), 281–316.

    Google Scholar 

  51. Yu.I. Karlovich and I.M. Spitkovsky, Almost periodic factorization: An analogue of Chebotarev’s algorithm, Contemporary Math. 189 (1995), 327–352.

    Google Scholar 

  52. Yu.I. Karlovich and I.M. Spitkovsky, Factorization of almost periodic matrix functions, J. Math. Anal. Appl. 193 (1995), 209–232.

    Google Scholar 

  53. Yu.I. Karlovich and I.M. Spitkovsky, Almost periodic polynomial factorization of some triangular matrix functions, Oper. Theory Adv. Appl. 202 (2010), 337–354.

    Google Scholar 

  54. Yu.I. Karlovich, I.M. Spitkovsky, and R.A. Walker, Almost periodic factorization of block triangular matrix functions revisited, Linear Algebra Appl. 293 (1999), 199–232.

    Article  MathSciNet  MATH  Google Scholar 

  55. V. Kokilashvili and S. Samko, Singular integrals in weighted Lebesgue spaces with variable exponent, Georgian Math. J. 10 (2003), 145–156.

    MathSciNet  MATH  Google Scholar 

  56. N.Ya. Krupnik, A criterion for singular integral operators with measurable coefficients to be Noetherian, Sakharth. SSR Mecn. Akad. Moambe 80 (1975), 533–536.

    MathSciNet  MATH  Google Scholar 

  57. G.S. Litvinchuk and I.M. Spitkovsky, Factorization of measurable matrix functions, Birkhäuser Verlag, Basel and Boston 1987.

    Google Scholar 

  58. C.V.M. van der Mee, L. Rodman, and I.M. Spitkovsky, Factorization of block triangular matrix functions with off diagonal binomials, Oper. Theory Adv. Appl. 160 (2005), 423–437.

    Google Scholar 

  59. C.V.M. van der Mee, L. Rodman, I.M. Spitkovsky, and H.J. Woerdeman, Factorization of block triangular matrix functions in Wiener algebras on ordered abelian groups, Oper. Theory Adv. Appl. 149 (2004), 441–465.

    Google Scholar 

  60. S.G. Mikhlin and S. Prössdorf, Singular integral operators, Springer-Verlag, Berlin 1986.

    Google Scholar 

  61. N.I. Muskhelishvili, Singular integral equations, Dover Publications Inc., New York 1992.

    Google Scholar 

  62. A.M. Nikolaichuk and I.M. Spitkovsky, The Riemann boundary value problem with a Hermitian matrix, Soviet Math. Dokl. 16 (1975), 533–536.

    Google Scholar 

  63. V. Pellegrini, General Wiener-Hopf operators and the numerical range of an operator, Proc. Amer. Math. Soc. 38 (1973), 141–146.

    Article  MathSciNet  MATH  Google Scholar 

  64. H.R. Pousson, Systems of Toeplitz operators on 𝐻2, Proc. Amer. Math. Soc. 19 (1968), 603–608.

    MathSciNet  MATH  Google Scholar 

  65. D. Quint, L. Rodman, and I.M. Spitkovsky, New cases of almost periodic factorization of triangular matrix functions, Michigan Math. J. 45 (1998), 73–102.

    Article  MathSciNet  MATH  Google Scholar 

  66. M. Rabindranathan, On the inversion of Toeplitz operators, J. Math. Mech. 19 (1969/1970), 195–206.

    Google Scholar 

  67. V. Rabinovich and S. Samko, Pseudodifferential operators approach to singular integral operators in weighted variable exponent Lebesgue spaces on Carleson curves, Integral Equations and Operator Theory 69 (2011), 405–444.

    Article  MathSciNet  MATH  Google Scholar 

  68. A. Rastogi, L. Rodman, and I.M. Spitkovsky, Almost periodic factorization of 2 × 2 matrix functions: New cases of off diagonal spectrum, 202 (2010), 469–487.

    Google Scholar 

  69. L. Rodman and I.M. Spitkovsky, Almost periodic factorization and corona theorem, Indiana Univ. Math. J. 47 (1998), 1243–1256.

    Article  MathSciNet  MATH  Google Scholar 

  70. , Factorization of matrix functions with subgroup supported Fourier coefficients, J. Math. Anal. Appl. 323 (2006), 604–613.

    Google Scholar 

  71. L. Rodman, I.M. Spitkovsky, and H.J. Woerdeman, Carath´eodory-Toeplitz and Nehari problems for matrix-valued almost periodic functions, Trans. Amer. Math. Soc. 350 (1998), 2185–2227.

    Article  MathSciNet  MATH  Google Scholar 

  72. L. Rodman, I.M. Spitkovsky, and H.J. Woerdeman, Factorization of almost periodic matrix functions of several variables and Toeplitz operators, Oper Theory. Adv. Appl. 122 (2001), 385–416.

    Google Scholar 

  73. L. Rodman, I.M. Spitkovsky, and H.J. Woerdeman, Fredholmness and invertibility of Toeplitz operators with matrix almost periodic symbols, Proc. Amer. Math. Soc. 130 (2002), 1365–1370.

    Google Scholar 

  74. L. Rodman, I.M. Spitkovsky, and H.J. Woerdeman, Noncanonical factorization of almost periodic multivariable matrix functions, Oper. Theory Adv. Appl. 142 (2003), 311–344.

    Google Scholar 

  75. W. Rudin, Fourier analysis on groups, John Wiley & Sons Inc., New York 1990.

    Google Scholar 

  76. D. Sarason, Toeplitz operators with piecewise quasicontinuous symbols, Indiana Univ. Math. J. 26 (1977), 817–838.

    Article  MathSciNet  MATH  Google Scholar 

  77. D. Sarason, Toeplitz operators with semi-almost periodic symbols, Duke Math. J. 44 (1977), no. 2, 357–364.

    Google Scholar 

  78. I.B. Simonenko, Riemann’s boundary problem with a measurable coefficient, Soviet Math. Dokl. 1 (1960), 1295–1298.

    MathSciNet  MATH  Google Scholar 

  79. I.B. Simonenko, The Riemann boundary value problem for 𝑛 pairs of functions with measurable coefficients and its application to the investigation of singular integrals in the spaces 𝐿𝑝 with weight, Izv. Akad. Nauk SSSR. Ser. Mat. 28 (1964), 277–306.

    Google Scholar 

  80. I.B. Simonenko, Some general questions of the theory of the Riemann boundary value problem, Izv. Akad. Nauk SSSR. Ser. Mat. 2 (1968), 1091–1099.

    Google Scholar 

  81. F.-O. Speck, General Wiener-Hopf factorization methods, Pitman, Boston 1985.

    Google Scholar 

  82. I.M. Spitkovskiǐ, Singular integral operators with a piecewise sectorial matrix symbol, Soobshch. Akad. Nauk Gruzin. SSR 121 (1986), 249–252.

    Google Scholar 

  83. I.M. Spitkovsky, On the factorization of almost periodic matrix functions, Math. Notes 45 (1989), no. 5–6, 482–488.

    Article  MathSciNet  Google Scholar 

  84. , Singular integral operators with 𝑃𝐶 symbols on the spaces with general weights, J. Functional Analysis 105 (1992), 129–143.

    Google Scholar 

  85. N.P. Vekua, Systems of singular integral equations, P. Noordhoff Ltd., Groningen 1967.

    Google Scholar 

  86. H. Widom, Inversion of Toeplitz matrices. III, Notices Amer. Math. Soc. 7 (1960), 63.

    Google Scholar 

  87. H. Widom, Singular integral equations in 𝐿𝑝, Trans. Amer. Math. Soc. 97 (1960), 131– 160.

    Google Scholar 

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Authors and Affiliations

  1. Fakultät für Mathematik, TU Chemnitz, D-09107, Chemnitz, Germany

    Albrecht Böttcher

  2. Department of Mathematics, College of William and Mary, Williamsburg, VA, 23187, USA

    Ilya M. Spitkovsky

Authors
  1. Albrecht Böttcher
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  2. Ilya M. Spitkovsky
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Correspondence to Albrecht Böttcher .

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Editors and Affiliations

  1. , Departamento de Matemática, Universidade de Aveiro, Campus Universitário de Santiago, Aveiro, 3810-193, Portugal

    Alexandre Almeida

  2. , Departamento de Matemática, Universidade de Aveiro, Campus Universitário de Santiago, Aveiro, 3810-193, Portugal

    Luís Castro

  3. Instituto Superior Tecnico, Depto. Matematica, Universidade Tecnica Lisboa, Av. Rovisco Pais 1, Lisboa, 1049-001, Portugal

    Frank-Olme Speck

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For Stefan Samko on his 70th birthday

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Böttcher, A., Spitkovsky, I.M. (2013). The Factorization Problem: Some Known Results and Open Questions. In: Almeida, A., Castro, L., Speck, FO. (eds) Advances in Harmonic Analysis and Operator Theory. Operator Theory: Advances and Applications, vol 229. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0516-2_6

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