Skip to main content
SpringerLink
Log in

Noncanonical Factorizations of Almost Periodic Multivariable Matrix Functions

  • Conference paper
Singular Integral Operators, Factorization and Applications

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 142))

Abstract

Noncanonical factorizations of almost periodic matrix-valued functions of several real variables are studied. In particular, results are proved concerning hereditary properties of such factorizations, symmetric factorization of matrix functions that possess certain symmetries, behavior of factorization indices under small perturbations, connections between left and right indices, and relations between factorization and Fredholmness properties of the associated Toeplitz operators. The last section is devoted to uses of factorization for normalization of bases, an important problem in wavelets and other applications. Conjectures and open problems are stated.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bakonyi, M., Rodman, L., Spitkovsky, I.M., and Woerdeman, H.J., Positive extensions of matrix functions of two variables with support in an infinite bandC.R. Acad. Sci. Paris Sér. I Math.323(8) (1996), 859–863.

    MathSciNet  MATH  Google Scholar 

  2. Bakonyi, M., Rodman, L., Spitkovsky, I.M., and Woerdeman, H.J., Positive matrix functions on the bitorus with prescribed coefficients in a bandJ. Fourier Analysis and Applications5 (1999), 789–812.

    MathSciNet  Google Scholar 

  3. Ball, J.A., Karlovich, Yu.I., Rodman, L., and Spitkovsky, I.M., Sarason interpolation and Toeplitz corona theorem for almost periodic matrix functionsIntegral Equations and Operator Theory32 (1998), 243–281.

    Article  MathSciNet  MATH  Google Scholar 

  4. Bastos, M.A., Karlovich, Yu.I., dos Santos, F.A., and Tishin, P.M., The Corona theorem and the existence of canonical factorization of triangular AP-matrix functionsJ. Math. Anal. Appl.223 (1998), 494–522.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Bastos, M.A., Karlovich, Yu.I., Spitkovsky, I.M., and Tishin, P.M., On a new algorithm for almost periodic factorization, OperatorTheory: Advances and Applications103 (1998), 53–74.

    MathSciNet  Google Scholar 

  7. Böttcher, A., Karlovich, Yu.I, and Spitkovsky, I.M.Convolution Operators and Factorization of Almost Periodic Matrix FunctionsBirkhäuser, Basel and Boston, 2002.

    Book  MATH  Google Scholar 

  8. Clancey, K.F., and Gohberg, I.Factorization of Matrix Functions and Singular Integral OperatorsBirkhäuser, Basel and Boston, 1981.

    MATH  Google Scholar 

  9. Coburn, L.A., and Douglas, R.G., Translation operators on the half-lineProc. Nat. Acad. Sci. U.S.A.62 (1969), 1010–1013.

    Article  MathSciNet  MATH  Google Scholar 

  10. Corduneanu, C.Almost Periodic FunctionsJ. Wiley & Sons, 1968.

    Google Scholar 

  11. Ehrhardt, T., Invertibility theory for Toeplitz Hankel operators and singular in-tegral operators with flip, preprint.

    Google Scholar 

  12. Feldman, I. and Markus, A., On some properties of factorization indicesIntegral Equations and Operator Theory30 (1998), 326–337.

    Article  MathSciNet  MATH  Google Scholar 

  13. Gohberg, I.C., and Fel’dman, I.A., Integro-difference Wiener-Hopf equationsActa Sci. Math. (Szeged)30 (1969), 199–224. (Russian.)

    MathSciNet  Google Scholar 

  14. Gohberg, I., and Feldman, I.Convolution Equations and Projection Methods for their SolutionNauka, Moscow, 1971. English translation Amer.Math. Soc. Transl. of Math. Monographs 41, Providence, R.I. 1974.

    Google Scholar 

  15. Gohberg, I., Goldberg, S., and Kaashoek, M.A.Classes of Linear Operators. IIBirkhäuser, Basel and Boston, 1993.

    Google Scholar 

  16. Gohberg, I., and Kaashoek, M.A. (eds.).Constructive Methods of Wiener-Hopf InterpolationBirkhäuser, Basel and Boston, 1986.

    Google Scholar 

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

    MathSciNet  Google Scholar 

  18. Gohberg, I., and Krupnik, N.Ya.One-Dimensional Linear Singular Integral Equations.; Vol. I: Introduction; Vol. 2: General Theory and ApplicationsBirkhäuser, Basel and Boston, 1992.

    Book  Google Scholar 

  19. Goodman, T.N.T., Micchelli, C.A., Rodriguez, G., and Seatzu, S. On the Cholesky factorization of the Gram matrix of multivariate functionsSIAM J. Matrix Anal. Appl.22 (2000), 501–526.

    Article  MathSciNet  MATH  Google Scholar 

  20. Horn, R.A., and Johnson, C.R.Matrix AnalysisCambridge University Press, 1985.

    Google Scholar 

  21. Karlovich, Yu.I.Algebras of Convolution Type Operators with Discrete Groups of Shifts and Oscillating CoefficientsDoctoral Dissertation, Mathematical Institute, Georgian Academy of Sciences, Tbilisi, 1991.

    Google Scholar 

  22. Karlovich, Yu.I., On the Baseman problemDemonstratio Math.26 (1993), 581–595.

    MathSciNet  MATH  Google Scholar 

  23. Karlovich, Yu.I., and Spitkovsky, I.M.Factorization of Almost periodic Matrix Functions and (semi) Fredholmness of Some Convolution Type EquationsVINITI, Moscow, 1985.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Karlovich, Yu.I., and Spitkovsky, I.M., (Semi)-Fredholmness of convolution operators on the spaces of Bessel potentialsOperator Theory: Advances and Applications71 (1994), 122–152.

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  28. Karlovich, Yu.I., and Spitkovsky, I.M., Semi-Fredholm properties of certain singular integral operatorsOperator Theory: Advances and Applications90 (1996), 264–287.

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  30. Krein, M.G., Integral equations on a half-line with kernel depending upon the difference of the arguments, Amer.Math. Soc. Transl. Series 2,22 (1962), 163–288.

    MathSciNet  Google Scholar 

  31. Levitan, B.M.Almost Periodic FunctionsGITTL, Moscow, 1953. (Russian.)

    Google Scholar 

  32. Levitan, B.M., and Zhikov, V.V.Almost Periodic Functions and Differential EquationsCambridge University Press, 1982.

    Google Scholar 

  33. Litvinchuk, L.S., and Spitkovsky, I.M.Factorization of Measurable Matrix FunctionsBirkhäuser Verlag, Basel and Boston, 1987.

    Google Scholar 

  34. Marshall, A.W. and Olkin, I.Inequalities: Theory of Majorization and its ApplicationsAcademic Press, New York-London, 1979.

    Google Scholar 

  35. van der Mee, C.V.M., Rodriguez, G., and Seatzu, S., Spectral factorization of bi-infinite multi-index block Toeplitz matricesLinear Algebra Appl.343/344 (2002), 355–380.

    MathSciNet  Google Scholar 

  36. van der Mee, C.V.M., Rodriguez, G., and Seatzu, S., Spectral factorization of bi-infinite block Toeplitz matrices with applications, preprint.

    Google Scholar 

  37. Nikolaicuk, A.M., and Spitkovsky, I.M., On the Riemann boundary-value problem with Hermitian matrixSoviet Math. Doklady16 (1975), 1280–1283.

    Google Scholar 

  38. Pankov, A.A.Bounded and Almost Periodic Solutions of Nonlinear Differential Operator EquationsKluwer, Dordrecht/Boston/London, 1990.

    Book  MATH  Google Scholar 

  39. Perov, A.I., and Kibenko, A.V.,Atheorem on the argument of an almost periodic function of several variablesLitovskii Matematicheskii Sbornik, 7 (1967), 505–508 (Russian).

    MathSciNet  Google Scholar 

  40. Quint, D., Rodman, L., and Spitkovsky, I.M., New cases of almost periodic factorization of triangular matrix functionsMichigan Math. J.45(1) (1998), 73–102.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  42. Rodman, L., Spitkovsky, I.M., and Woerdeman, H.J., Carathéodory-Toeplitz and Nehari problems for matrix-valued almost periodic functionsTrans. Amer. Math. Soc.350 (1998), 2185–2227.

    Article  MathSciNet  MATH  Google Scholar 

  43. Rodman, L., Spitkovsky, I.M., and Woerdeman, H.J., Factorization of almost periodic matrix functions of several variables and Toeplitz operatorsOperator Theory: Advances and Applications122 (2001), (H. Bart, I. Gohberg, A.C.M. Ran, eds.), 385–416.

    Google Scholar 

  44. Rodman, L., Spitkovsky, I.M., and Woerdeman, H. J., Multiblock problems for almost periodic matrix functions of several variablesNew York J. Math. 7(2001), 117–148.

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  46. Rodman, L., Spitkovsky, I.M., and Woerdeman, H.J., Contractive extension problems for matrix-valued almost periodic functions of several variablesJournal of Operator Theoryto appear.

    Google Scholar 

  47. Rodman, L., Spitkovsky, I.M., and Woerdeman, H.J., Abstract band method via factorization, positive and band extensions of multivariable almost periodic matrix functions, and spectral estimationMemoirs of the AMSto appear.

    Google Scholar 

  48. Spitkovsky, I.M., On the factorization of almost periodic functions.Math. Notes45 (1989), 482–488.

    Article  MathSciNet  Google Scholar 

  49. Spitkovsky, I.M., and Woerdeman, H.J., The Carath¨¦odory-Toeplitz problem for almost periodic functionsJ. Functional Analysis115(2) (1993), 281–293.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, College of William and Mary Williamsburg, VA 23187-8795, USA

    Leiba Rodman, Ilya M. Spitkovsky & Hugo J. Woerdeman

Authors
  1. Leiba Rodman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ilya M. Spitkovsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hugo J. Woerdeman
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Faculty of Mathematics, Technical University Chemnitz, Chemnitz, 09107, Germany

    Albrecht Böttcher

  2. Department of Mathematics and Computer Science, Vrije Universiteit Amsterdam, De Boelelaan, Amsterdam, NL-1081 HV, The Netherlands

    Marinus A. Kaashoek

  3. Departamento de Matemática, Instituto Superior Técnico, Ave. Rovisco Pais, Lisboa, 1049-001, Portugal

    Amarino Brites Lebre , António Ferreira dos Santos  & Frank-Olme Speck ,  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer Basel AG

About this paper

Cite this paper

Rodman, L., Spitkovsky, I.M., Woerdeman, H.J. (2003). Noncanonical Factorizations of Almost Periodic Multivariable Matrix Functions. In: Böttcher, A., Kaashoek, M.A., Lebre, A.B., dos Santos, A.F., Speck, FO. (eds) Singular Integral Operators, Factorization and Applications. Operator Theory: Advances and Applications, vol 142. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8007-7_17

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-8007-7_17

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9401-2

  • Online ISBN: 978-3-0348-8007-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation