Skip to main content
SpringerLink
Log in

Factorization Versus Invertibility of Matrix Functions on Compact Abelian Groups

  • Chapter
  • First Online:
A Panorama of Modern Operator Theory and Related Topics

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 218))

  • 970 Accesses

Abstract

Open problems are stated and some new results are proved concerning the relationship between invertibility and factorization in various Banach algebras of matrix-valued functions on connected compact abelian groups.

Mathematics Subject Classification (2000). 47A56, 47A68.

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. 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), no. 2, 143–156.

    Google Scholar 

  2. 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 

  3. H. Bart, I. Gohberg, and M.A. Kaashoek, Invariants for Wiener-Hopf equivalence of analytic operator functions, Constructive methods of Wiener-Hopf factorization, Operator Theory: Advances and Applications, vol. 21, Birkhäuser, Basel, 1986, pp. 317–355.

    Google Scholar 

  4. H. Bart, I. Gohberg, M.A. Kaashoek, and A.C.M. Ran, A state space approach to canonical factorization with applications, OT vol. 200, Birkhäuser Verlag, Basel and Boston, 2010.

    Google Scholar 

  5. M.A. Bastos, Yu.I. Karlovich, I.M. Spitkovsky, and P.M. Tishin, On a new algorithm for almost periodic factorization, Recent Progress in Operator Theory (Regensburg, 1995) (I. Gohberg, R. Mennicken, and C. Tretter, eds.), Operator Theory: Advances and Applications, vol. 103, Birkhäuser Verlag, Basel and Boston, 1998, pp. 53–74.

    Google Scholar 

  6. 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 

  7. A. Böttcher, Yu.I. Karlovich, and I.M. Spitkovsky, Convolution operators and factorization of almost periodic matrix functions, OT vol. 131, Birkhäuser Verlag, Basel and Boston, 2002.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. A. Brudnyi, L. Rodman, and I.M. Spitkovsky, Projective free algebras of continuous functions on compact abelian groups, J. Functional Analysis 259 (2010), 918–932.

    Google Scholar 

  10. 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 

  11. 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 

  12. M.C. Câmara, C. Diogo, Yu.I. Karlovich, and I.M. Spitkovsky, Factorizations, Riemann-Hilbert problems and the corona theorem, arXiv:1103.1935v1 [math.FA] (2011), 1–32.

    Google Scholar 

  13. M.C. Câmara, Yu.I. Karlovich, and I.M. Spitkovsky, Almost periodic factorization of some triangular matrix functions, Modern Analysis and Applications. The Mark Krein Centenary Conference (V. Adamyan, Y. Berezansky, I. Gohberg, M. Gorbachuk, A. Kochubei, H. Langer, and G. Popov, eds.), Operator Theory: Advances and Applications, vol. 190, Birkhäuser Verlag, Basel and Boston, 2009, pp. 171–190.

    Google Scholar 

  14. K.F. Clancey and I. Gohberg, Factorization of matrix functions and singular integral operators, OT vol. 3, Birkhäuser, Basel and Boston, 1981.

    Google Scholar 

  15. L. Coburn and R.G. Douglas, Translation operators on the half-line, Proc. Nat. Acad. Sci. USA 62 (1969), 1010–1013.

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  17. H.G. Dales, Banach algebras and automatic continuity, London Mathematical Society Monographs. New Series, vol. 24, The Clarendon Press Oxford University Press, New York, 2000, Oxford Science Publications.

    Google Scholar 

  18. T.W. Dawson and J.F. Feinstein, On the denseness of the invertible group in Banach algebras, Proc. Amer. Math. Soc. 131 (2003), no. 9, 2831–2839.

    Article  MathSciNet  MATH  Google Scholar 

  19. T. Ehrhardt and C.V.M. van der Mee, Canonical factorization of continuous functions on the d-torus, Proc. Amer. Math. Soc. 131 (2003), no. 3, 801–813.

    Article  MathSciNet  MATH  Google Scholar 

  20. 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 

  21. M.P. Ganin, On a Fredholm integral equation whose kernel depends on the difference of the arguments, Izv. Vys. Uchebn. Zaved. Matematika (in Russian) (1963), no. 2 (33), 31–43.

    Google Scholar 

  22. 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 

  23. I. Gohberg, S. Goldberg, and M.A. Kaashoek, Classes of linear operators. Vol. II, Birkhäuser Verlag, Basel and Boston, 1993.

    Google Scholar 

  24. I. Gohberg, M.A. Kaashoek, and I.M. Spitkovsky, An overview of matrix factorization theory and operator applications, Operator Theory: Advances and Applications 141 (2003), 1–102.

    MathSciNet  Google Scholar 

  25. I. Gohberg, M.A. Kaashoek, and I.M. Spitkovsky, Similarity of operator blocks and canonical forms. II. Infinite-dimensional case and Wiener-Hopf factorization, Topics in modern operator theory (Timi¸soara/Herculane, 1980), Operator Theory: Advances and Applications, vol. 2, Birkhäuser, Basel, 1981, pp. 121–170.

    Google Scholar 

  26. I. Gohberg, M.A. Kaashoek, and F. van Schagen, Partially specified matrices and operators: classification, completion, applications, OT vol. 79, Birkhäuser Verlag, Basel, 1995.

    Google Scholar 

  27. I. Gohberg and M.G. Krein, Systems of integral equations on a half-line with kernel depending upon the difference of the arguments, Uspekhi Mat. Nauk 13 (1958), no. 2, 3–72 (in Russian), English translation: Amer. Math. Soc. Transl. 14 (1960), no. 2, 217–287.

    Google Scholar 

  28. I. Gohberg and N. Krupnik, One-dimensional linear singular integral equations. Introduction, OT 53, 54, vol. 1 and 2, Birkhäuser Verlag, Basel and Boston, 1992.

    Google Scholar 

  29. I.C. Gohberg and I.A. Feldman, Integro-difference Wiener-Hopf equations, Acta Sci. Math. Szeged 30 (1969), no. 3–4, 199–224 (in Russian).

    Google Scholar 

  30. I.C. Gohberg and I.A. Feldman, Convolution equations and projection methods for their solution, Nauka, Moscow, 1971 (in Russian), English translation Amer. Math. Soc. Transl. of Math. Monographs 41, Providence, R.I. 1974.

    Google Scholar 

  31. Yu.I. Karlovich and I.M. Spitkovsky, Factorization of almost periodic matrix functions and (semi) Fredholmness of some convolution type equations, No. 4421-85 dep., VINITI, Moscow, 1985, in Russian.

    Google Scholar 

  32. Yu.I. Karlovich and I.M. Spitkovsky, On the theory of systems of equations of convolution type with semi-almostperiodic symbols in spaces of Bessel potentials, SovietMath.Dokl. 33 (1986), 145–149.

    Google Scholar 

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

    Google Scholar 

  34. G.S. Litvinchuk and I.M. Spitkovsky, Factorization of measurable matrix functions, OT vol. 25, Birkhäuser Verlag, Basel and Boston, 1987.

    Google Scholar 

  35. C.V.M. van der Mee, L. Rodman, and I.M. Spitkovsky, Factorization of block triangular matrix functions with off diagonal binomials, Operator Theory: Advances and Applications 160 (2005), 423–437.

    Google Scholar 

  36. 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, Operator Theory: Advances and Applications 149 (2004), 441–465.

    Google Scholar 

  37. B.V. Pal’cev, Convolution equations on a finite interval for a class of symbols having power asymptotics at infinity, Izv. Akad. Nauk SSSR. Mat. 44 (1980), 322–394 (in Russian), English translation: Math. USSR Izv. 16 (1981).

    Google Scholar 

  38. B.V. Pal’cev, A generalization of the Wiener-Hopf method for convolution equations on a finite interval with symbols having power asymptotics at infinity, Mat. Sb. 113 (155) (1980), 355–399 (in Russian), English translation: Math. USSR Sb. 41 (1982).

    Google Scholar 

  39. A.R. Pears, Dimension theory of general spaces, Cambridge University Press, Cambridge, England, 1975.

    Google Scholar 

  40. A. Rastogi, L. Rodman, and I.M. Spitkovsky, Almost periodic factorization of 2 × 2 matrix functions: New cases of off diagonal spectrum, Recent Advances and New Directions in Applied and Pure Operator Theory (Williamsburg, 2008) (J.A. Ball, V. Bolotnikov, J.W. Helton, L. Rodman, and I.M. Spitkovsky, eds.), Operator Theory: Advances and Applications, vol. 202, Birkhäuser, Basel, 2010, pp. 469–487.

    Google Scholar 

  41. G. Robertson, On the density of the invertible group in C ∗ -algebras, Proc. Edinburgh Math. Soc. (2) 20 (1976), no. 2, 153–157.

    Google Scholar 

  42. L. Rodman and I.M. Spitkovsky, Factorization of matrix functions with subgroup supported Fourier coefficients, J. Math. Anal. Appl. 323 (2006), 604–613.

    Article  MathSciNet  MATH  Google Scholar 

  43. W. Rudin, Fourier analysis on groups, John Wiley & Sons Inc., New York, 1990, Reprint of the 1962 original, a Wiley-Interscience Publication.

    Google Scholar 

  44. I.M. Spitkovsky, Factorization of several classes of semi-almost periodic matrix functions and applications to systems of convolution equations, Izvestiya VUZ., Mat. (1983), no. 4, 88–94 (in Russian), English translation in Soviet Math. – Iz. VUZ 27 (1983), 383–388.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Calgary, 2500 University Dr. NW, Calgary, Alberta, Canada, T2N 1N4

    Alex Brudnyi

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

    Leiba Rodman & Ilya M. Spitkovsky

Authors
  1. Alex Brudnyi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Leiba Rodman
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ilya M. Spitkovsky
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alex Brudnyi .

Editor information

Editors and Affiliations

  1. Dept. Mathematics, Weizmann Institute of Science, Rehovot, Israel

    Harry Dym

  2. Dept. Mathematics &, Computer Sciences, Free University Amsterdam, De Boelelaan 1081 A, Amsterdam, 1081 HV, Netherlands

    Marinus A. Kaashoek

  3. Dept. Mathematics & Statistics, University of Calgary, University Drive NW 2500, Calgary, T2N 1N4, Alberta, Canada

    Peter Lancaster

  4. Inst. Analysis und Scientific Computing, TU Wien, Wiedner Hauptstr. 8-10, Wien, 1040, Austria

    Heinz Langer

  5. Dept. Mathematics, Technion Israel Institute of Technology, Haifa, 32000, Israel

    Leonid Lerer

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Basel AG

About this chapter

Cite this chapter

Brudnyi, A., Rodman, L., Spitkovsky, I.M. (2012). Factorization Versus Invertibility of Matrix Functions on Compact Abelian Groups. In: Dym, H., Kaashoek, M., Lancaster, P., Langer, H., Lerer, L. (eds) A Panorama of Modern Operator Theory and Related Topics. Operator Theory: Advances and Applications(), vol 218. Springer, Basel. https://doi.org/10.1007/978-3-0348-0221-5_9

Download citation

Publish with us

Policies and ethics

Search

Navigation