The Operator Fejér-Riesz Theorem

  • Michael A. Dritschel
  • James Rovnyak
Part of the Operator Theory Advances and Applications book series (OT, volume 207)


The Fejér-Riesz theorem has inspired numerous generalizations in one and several variables, and for matrix- and operator-valued functions. This paper is a survey of some old and recent topics that center around Rosenblum’s operator generalization of the classical Fejér-Riesz theorem.

Mathematics Subject Classification (2000)

Primary 47A68 Secondary 60G25 47A56 47B35 42A05 32A70 30E99 


Trigonometric polynomial Fejér-Riesz theorem spectral factorization Schur complement noncommutative polynomial Toeplitz operator shift operator 


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  1. [1]
    D.Z. Arov, Stable dissipative linear stationary dynamical scattering systems, J. Operator Theory 2 (1979), no. 1, 95–126, English Transl, with appendices by the author and J. Rovnyak, Oper. Theory Adv. Appl., vol. 134, Birkhäuser Verlag, Basel, 2002, 99–136.zbMATHMathSciNetGoogle Scholar
  2. [2]
    M. Bakonyi and T. Constantinescu, Schur’s algorithm and several applications, Pitman Research Notes in Mathematics Series, vol. 261, Longman Scientific & Technical, Harlow, 1992.Google Scholar
  3. [3]
    M. Bakonyi and H.J. Woerdeman, Matrix completions, moments, and sums of Hermitian squares, book manuscript, in preparation, 2008.Google Scholar
  4. [4]
    S. Barclay, A solution to the Douglas-Rudin problem for matrix-valued functions, Proc. London Math. Soc. (3), 99 (2009), no. 3, 757–786.Google Scholar
  5. [5]S. Barclay, Continuity of the spectral factorization mapping, J. London Math. Soc. (2) 70 (2004), no. 3, 763–779.Google Scholar
  6. [6] S. Barclay, Banach spaces of analytic vector-valued functions, Ph.D. thesis, University of Leeds, 2007.Google Scholar
  7. [7]
    H. Bart, I. Gohberg, M.A. Kaashoek, and A.C.M. Ran, Factorization of matrix and operator functions: the state space method, Oper. Theory Adv. Appl., vol. 178, Birkhäuser Verlag, Basel, 2008.Google Scholar
  8. [8]
    G. Blower, On analytic factorization of positive Hermitian matrix functions over the bidisc, Linear Algebra Appl. 295 (1999), no. 1-3, 149–158.CrossRefzbMATHMathSciNetGoogle Scholar
  9. [9]
    J. Bourgain, A problem of Douglas and Rudin on factorization, Pacific J. Math. 121 (1986), no. 1, 47–50.zbMATHMathSciNetGoogle Scholar
  10. [10]
    L. de Branges, The expansion theorem for Hilbert spaces of entire functions, Entire Functions and Related Parts of Analysis (Proc. Sympos. Pure Math., La Jolla, Calif., 1966), Amer. Math. Soc, Providence, RI, 1968, pp. 79–148.Google Scholar
  11. [11]
    A. Brown and P.R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213 (1963/1964), 89–102.Google Scholar
  12. [12]
    G. Cassier, Problème des moments sur un compact de Rn et décomposition de polynômes à plusieurs variables, J. Funct. Anal. 58 (1984), no. 3, 254–266.CrossRefzbMATHMathSciNetGoogle Scholar
  13. [13]
    K.F. Clancey and I. Gohberg, Factorization of matrix functions and singular integral operators, Oper. Theory Adv. Appl., vol. 3, Birkhäuser Verlag, Basel, 1981.Google Scholar
  14. [14]
    T. Constantinescu, Factorization of positive-definite kernels, Topics in operator theory: Ernst D. Hellinger memorial volume, Oper. Theory Adv. Appl., vol. 48, Birkhäuser Verlag, Basel, 1990, pp. 245–260.Google Scholar
  15. [15]
    A. Devinatz, The factorization of operator-valued functions, Ann. of Math. (2) 73 (1961), 458–495.Google Scholar
  16. [16]
    R.G. Douglas, On factoring positive operator functions, J. Math. Mech. 16 (1966), 119–126.zbMATHMathSciNetGoogle Scholar
  17. [17]
    R.G. Douglas and W. Rudin, Approximation by inner functions, Pacific J. Math. 31 (1969), 313–320.zbMATHMathSciNetGoogle Scholar
  18. [18]
    M.A. Dritschel, On factorization of trigonometric polynomials, Integral Equations Operator Theory 49 (2004), no. 1, 11–42.CrossRefzbMATHMathSciNetGoogle Scholar
  19. [19]
    M.A. Dritschel and H.J. Woerdeman, Outer factorizations in one and several variables, Trans. Amer. Math. Soc. 357 (2005), no. 11, 4661–4679.CrossRefzbMATHMathSciNetGoogle Scholar
  20. [20]
    P.L. Duren, Theory of H v spaces, Academic Press, New York, 1970; Dover reprint, Mineóla, New York, 2000.zbMATHGoogle Scholar
  21. [21]
    R.E. Edwards, Functional analysis. Theory and applications, Holt, Rinehart and Winston, New York, 1965; Dover reprint, Mineóla, New York, 1995.zbMATHGoogle Scholar
  22. [22]
    L. Fejér, Über trigonometrische Polynome, J. Reine Angew. Math. 146 (1916), 53–82.CrossRefGoogle Scholar
  23. [23]
    J.S. Gerónimo and Ming-Jun Lai, Factorization of multivariate positive Laurent polynomials, J. Approx. Theory 139 (2006), no. 1–2, 327–345.CrossRefzbMATHMathSciNetGoogle Scholar
  24. [24]
    J.S. Geronimo and H.J. Woerdeman, Positive extensions, Fejér-Riesz factorization and autoregressive filters in two variables, Ann. of Math. (2) 160 (2004), no. 3, 839–906.Google Scholar
  25. [25]H.J. Woerdeman, The operator-valued autoregressive filter problem and the suboptimal Nehari problem in two variables, Integral Equations Operator Theory 53 (2005), no. 3, 343–361.CrossRefzbMATHMathSciNetGoogle Scholar
  26. [26]
    I. Gohberg, The factorization problem for operator functions, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 1055–1082, Amer. Math. Soc. Transl. (2) 49 130–161.zbMATHMathSciNetGoogle Scholar
  27. [27]
    I. Gohberg, S. Goldberg, and M.A. Kaashoek, Classes of linear operators. Vol. I, Oper. Theory Adv. Appl., vol. 49, Birkhäuser Verlag, Basel, 1990.Google Scholar
  28. [28]
    U. Grenander and G. Szegő, Toeplitz forms and their applications, California Monographs in Mathematical Sciences, University of California Press, Berkeley, 1958.Google Scholar
  29. [29]
    P.R. Halmos, Shifts on Hilbert spaces, J. Reine Angew. Math. 208 (1961), 102–112.CrossRefzbMATHMathSciNetGoogle Scholar
  30. [30]
    H. Helson, Lectures on invariant subspaces, Academic Press, New York, 1964.zbMATHGoogle Scholar
  31. [31]
    H. Helson and D. Lowdenslager, Prediction theory and Fourier series in several variables, Acta Math. 99 (1958), 165–202.CrossRefzbMATHMathSciNetGoogle Scholar
  32. [32]D. Lowdenslager, Prediction theory and Fourier series in several variables. II, Acta Math. 106 (1961), 175–213.CrossRefzbMATHMathSciNetGoogle Scholar
  33. [33]
    J.W. Helton, S.A. McCullough, and M. Putinar, Matrix representations for positive noncommutative polynomials, Positivity 10 (2006), no. 1, 145–163.CrossRefzbMATHMathSciNetGoogle Scholar
  34. [34]
    J.W. Helton and M. Putinar, Positive polynomials in scalar and matrix variables, the spectral theorem, and optimization, Operator theory, structured matrices, and dilations, Theta Ser. Adv. Math., vol. 7, Theta, Bucharest, 2007, pp. 229–306.Google Scholar
  35. [35]
    K. Hoffman, Banach spaces of analytic functions, Prentice-Hall Inc., Englewood Cliffs, N. J., 1962; Dover reprint, Mineóla, New York, 1988.zbMATHGoogle Scholar
  36. [36]
    R.B. Holmes, Geometric functional analysis and its applications, Graduate Texts in Mathematics, No. 24, Springer-Verlag, New York, 1975.Google Scholar
  37. [37]
    B. Jacob and J.R. Partington, On the boundedness and continuity of the spectral factorization mapping, SIAM J. Control Optim. 40 (2001), no. 1, 88–106.CrossRefzbMATHMathSciNetGoogle Scholar
  38. [38]
    T. Kailath, A.H. Sayed, and B. Hassibi, Linear estimation, Prentice Hall, Englewood Cliffs, NJ, 1980.Google Scholar
  39. [39]
    A. Lebow and M. Schreiber, Polynomials over groups and a theorem of Fejér and Riesz, Acta Sci. Math. (Szeged) 44 (1982), no. 3–4, 335–344 (1983).zbMATHMathSciNetGoogle Scholar
  40. [40]
    D. Lowdenslager, On factoring matrix-valued functions, Ann. of Math. (2) 78 (1963), 450–454.Google Scholar
  41. [41]
    A.S. Markus, Introduction to the spectral theory of polynomial operator pencils, Translations of Mathematical Monographs, vol. 71, Amer. Math. Soc., Providence, RI, 1988.Google Scholar
  42. [42]
    S. McCullough, Factorization of operator-valued polynomials in several non-commuting variables, Linear Algebra Appl. 326 (2001), no. 1–3, 193–203.CrossRefzbMATHMathSciNetGoogle Scholar
  43. [43]
    G.J. Murphy, C*-algebras and operator theory, Academic Press Inc., Boston, MA, 1990.Google Scholar
  44. [44]
    A. Naftalevich and M. Schreiber, Trigonometric polynomials and sums of squares, Number theory (New York, 1983-84), Lecture Notes in Math., vol. 1135, Springer-Verlag, Berlin, 1985, pp. 225–238.CrossRefGoogle Scholar
  45. [45]
    T.W. Palmer, Banach algebras and the general theory of *-algebras. Vol. 1, Encyclopedia of Mathematics and its Applications, vol. 49, Cambridge University Press, Cambridge, 1994.Google Scholar
  46. [46]T.W. Palmer, Banach algebras and the general theory of *-algebras. Vol. 2, Encyclopedia of Mathematics and its Applications, vol. 79, Cambridge University Press, Cambridge, 2001.Google Scholar
  47. [47]
    F. Riesz, Über ein Problem des Herrn Carathéodory, J. Reine Angew. Math. 146 (1916), 83–87.CrossRefGoogle Scholar
  48. [48]
    M. Rosenblatt, A multi-dimensional prediction problem, Ark. Mat. 3 (1958), 407–424.CrossRefzbMATHMathSciNetGoogle Scholar
  49. [49]
    M. Rosenblum, Vectorial Toeplitz operators and the Fejér-Riesz theorem, J. Math. Anal. Appl. 23 (1968), 139–147.CrossRefzbMATHMathSciNetGoogle Scholar
  50. [50]
    M. Rosenblum and J. Rovnyak, Hardy classes and operator theory, Oxford University Press, New York, 1985; Dover reprint, Mineóla, New York, 1997.zbMATHGoogle Scholar
  51. [51]J. Rovnyak, The factorization problem for nonnegative operator-valued functions, Bull. Amer. Math. Soc. 77 (1971), 287–318.CrossRefzbMATHMathSciNetGoogle Scholar
  52. [52]
    Yu.A. Rozanov, Stationary random processes, Holden-Day Inc., San Francisco, Calif., 1967.zbMATHGoogle Scholar
  53. [53]
    W. Rudin, The extension problem for positive-definite functions, Illinois J. Math. 7 (1963), 532–539.zbMATHMathSciNetGoogle Scholar
  54. [54]
    L.A. Sakhnovich, Interpolation theory and its applications, Kluwer, Dordrecht, 1997.zbMATHGoogle Scholar
  55. [55]
    D. Sarason, Generalized interpolation in H , Trans. Amer. Math. Soc. 127 (1967), 179–203.zbMATHMathSciNetGoogle Scholar
  56. [56]
    A.H. Sayed and T. Kailath, A survey of spectral factorization methods, Numer. Linear Algebra Appl. 8 (2001), no. 6–7, 467–496, Numerical linear algebra techniques for control and signal processing.CrossRefzbMATHMathSciNetGoogle Scholar
  57. [57]
    C. Scheiderer, Sums of squares of regular functions on real algebraic varieties, Trans. Amer. Math. Soc. 352 (2000), no. 3, 1039–1069.CrossRefzbMATHMathSciNetGoogle Scholar
  58. [58]
    K. Schmüdgen, The K-moment problem for compact semi-algebraic sets, Math. Ann. 289 (1991), no. 2, 203–206.CrossRefzbMATHMathSciNetGoogle Scholar
  59. [59]K. Schmüdgen, Noncommutative real algebraic geometry — some basic concepts and first ideas, Emerging Applications of Algebraic Geometry, The IMA Volumes in Mathematics and its Applications, vol. 149, Springer-Verlag, Berlin, 2009, pp. 325–350.Google Scholar
  60. [60]
    B. Simon, Orthogonal polynomials on the unit circle. Part 1, Amer. Math. Soc. Colloq. Publ., vol. 54, Amer. Math. Soc, Providence, RI, 2005.Google Scholar
  61. [61]
    B. Sz.-Nagy and C. Foias, Harmonic analysis of operators on Hilbert space, North-Holland Publishing Co., Amsterdam, 1970.Google Scholar
  62. [62]
    G. Szegő, Orthogonal polynomials, fourth ed., Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc, Providence, RI, 1975.Google Scholar
  63. [63]
    N. Wiener and P. Masani, The prediction theory of multivariate stochastic processes. I. The regularity condition, Acta Math. 98 (1957), 111–150.CrossRefzbMATHMathSciNetGoogle Scholar
  64. [64]P. Masani, The prediction theory of multivariate stochastic processes. II. The linear predictor, Acta Math. 99 (1958), 93–137.CrossRefMathSciNetGoogle Scholar
  65. [65]
    V. Zasuhin, On the theory of multidimensional stationary random processes, C. R. (Doklady) Acad. Sci. URSS (N.S.) 33 (1941), 435–437.MathSciNetGoogle Scholar

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© Springer Basel AG 2010

Authors and Affiliations

  • Michael A. Dritschel
    • 1
  • James Rovnyak
    • 2
  1. 1.School of Mathematics and Statistics Herschel BuildingUniversity of NewcastleNewcastle upon TyneUK
  2. 2.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA

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