Skip to main content
SpringerLink
Log in

Factorization of Matrices with Symmetries over Function Algebras

  • Published:
Integral Equations and Operator Theory Aims and scope Submit manuscript

Abstract

Factorizations of the Wiener–Hopf type of classes of matrix functions with various symmetries are studied, in the abstract context of Banach algebras of functions over connected abelian compact groups. The symmetries in question are induced by involutive automorphisms or antiautomorphisms of the general linear group, and include many symmetries studied previously in the literature. In the present paper the focus is on quasicanonical (i.e., with equal indices) and canonical factorizations.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aczél, J.: Lectures on Functional Equations and their Applications. In: Mathematics in Science and Engineering, vol. 19. Academic Press, New York (1966) (translated by Scripta Technica, Inc., supplemented by the author, edited by Hansjorg Oser)

  2. Artin M.: Algebra. Prentice Hall, London (1991)

    Google Scholar 

  3. Baer R.: Linear Algebra and Projective Geometry. Academic Press Inc., New York (1952)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. Baskakov A.G.: Asymptotic estimates for elements of matrices of inverse operators, and harmonic analysis. Sib. Math. J. 38, 10–22 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  6. Borel A., Tits J.: Théorèmes de structure et de conjugaison pour les groupes algébriques linéaires. C. R. Acad. Sci. Paris Sér. A–B 287(2), A55–A57 (1978)

    MathSciNet  Google Scholar 

  7. Brudnyi A., Rodman L., Spitkovsky I.M.: Non-denseness of factorable matrix functions. J. Funct. Anal. 261, 1969–1991 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  8. Corduneanu C.: Almost Periodic Functions. Wiley, New York (1968)

    MATH  Google Scholar 

  9. Cross G.W., Lancaster P.: Square roots of complex matrices. Linear Multilinear Algebra 1, 289–293 (1974)

    Article  MathSciNet  Google Scholar 

  10. Ehrhardt, T.: Factorization theory for Toeplitz + Hankel operators and singular integral operator with flip. Habilitation thesis, Technische Universität Chemnitz, Chemnitz (2004)

  11. Ehrhardt T.: Invertibility theory for Toeplitz plus Hankel operators and singular integral operators with flip. J. Funct. Anal. 208(1), 64–106 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Ehrhardt T., van der Mee C.V.M.: Canonical factorization of continuous functions on the d-torus. Proc. Am. Math. Soc. 131(3), 801–813 (2003)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  14. Ehrhardt, T., Spitkovsky, I.M.: On the kernel and cokernel of some Toeplitz operators. In: Operator Theory: Advances and Applications, vol. 237. Birkhäuser Verlag, Basel, pp. 127–144 (2013)

  15. Feldman I., Markus A.: On some properties of factorization indices. Integral Equ. Oper. Theory 30, 326–337 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  16. Gohberg I., Goldberg S., Kaashoek M.A.: Classes of Linear Operators (vol. II). Birkhäuser Verlag, Basel (1993)

    Book  MATH  Google Scholar 

  17. Gohberg, I., Krein, M.G.: Systems of integral equations on a half-line with kernel depending upon the difference of the arguments. Uspekhi Mat. Nauk 13(2), 3–72 (1958) (in Russian) (English translation: Am. Math. Soc. Transl. 14(2), 217–287, 1960)

  18. Gohberg, I.C., Feldman, I.A.:Convolution Equations and Projection Methods for Their solution. Nauka, Moscow (1971) (in Russian) (English translation: Am. Math. Soc. Transl. Math. Monogr. 41, Providence, 1974)

  19. Guralnick R.M., Li C.-K., Rodman L.: Multiplicative maps on invertible matrices that preserve matricial properties. Electron. J. Linear Algebra 10, 291–319 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  20. Hewitt E., Ross K.A.: Abstract Harmonic Analysis, vol. 1. Springer-Verlag, Berlin (1963)

    Book  Google Scholar 

  21. Higham N.J., Lin L.: On pth roots of stochastic matrices. Linear Algebra Appl. 435, 448–463 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  22. Horn R.A., Johnson C.R.: Matrix Analysis. Cambridge University Press, New York (1985)

    Book  MATH  Google Scholar 

  23. Kaashoek, M.A., Ran, A.C.M.: Symmetric Wiener–Hopf factorization of selfadjoint rational matrix functions and realization, constructive methods of Wiener–Hopf factorization. In: Operator Theory: Advances and Applications, vol. 21. Birkhäuser, Basel, pp. 373–409 (1986)

  24. Karlovich, Yu.I., Spitkovsky, I.M.: Factorization of Almost Periodic Matrix Functions and (Semi) Fredholmness of Some Convolution Type Equations No. 4421-85 Dep. VINITI, Moscow (1985) (in Russian)

  25. Kravchenko V.G., Lebre A.B., Rodríguez J.S.: Matrix functions consimilar to the identity and singular integral operators. Complex Anal. Oper. Theory 2(4), 593–615 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  26. Lancaster P., Rodman L.: Canonical forms for Hermitian matrix pairs under strict equivalence and congruence. SIAM Rev. 47, 407–443 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  27. Litvinchuk, G.S., Spitkovskii, I.M.: Factorization of Measurable Matrix Functions. In: Operator Theory: Advances and Applications, vol. 25. Birkhäuser Verlag, Basel (1987) (translated from the Russian by B. Luderer, with a foreword by B. Silbermann)

  28. MacLane, S., Birkhoff, G.: Algebra. Chelsea, Cambridge (1988)

  29. Marshall A.W., Olkin I., Arnold B.C.: Inequalities: theory of majorization and its applications, 2nd edn. Springer Series in Statistics. Springer, New York (2011)

    Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  32. Muirhead R.F.: Some methods applicable to identities and inequalities of symmetric algebraic functions of n letters. Proc. Edinb. Math. Soc. 21, 144–157 (1903)

    Article  Google Scholar 

  33. Nikolaichuk, A.M., Spitkovsky, I.M.: The Riemann boundary value problem with a Hermitian matrix, Dokl. Akad. Nauk SSSR 221(6), 1280–1283 (1975) (in Russian) (English translation: Sov. Math. Dokl. 16, 533–536, 1975)

  34. Psarrakos P.J.: On the mth roots of a complex matrix. Electron. J. Linear Algebra 9, 32–41 (2002)

    MATH  MathSciNet  Google Scholar 

  35. Rodman, L., Spitkovsky, I.M., Woerdeman, H.J.: Factorization of almost periodic matrix functions of several variables and Toeplitz operators. Operator theory and analysis (Amsterdam, 1997). In: Bart, H., Gohberg, I., Ran, A.C.M. (eds.) Operator Theory: Advances and Applications, vol. 122. Birkhäuser, Basel, pp. 385–416 (2001)

  36. Rudin, W.: Fourier Analysis on Groups. Wiley, New York (1990) (reprint of the 1962 original, A Wiley-Interscience Publication)

  37. Shmulyan, Yu.L.: The Riemann problem with a Hermitian matrix. Uspekhi Mat. Nauk 9(4), 243–248 (1954) (in Russian)

  38. Shubin M.A.: Factorization of matrix functions dependent on a parameter in normed rings, and related questions in the theory of Noetherian operators. Mat. Sb. (N.S.) 73((115), 610–629 (1967)

    MathSciNet  Google Scholar 

  39. Spitkovsky I.M.: On the factorization of almost periodic matrix functions. Math. Notes 45(5–6), 482–488 (1989)

    Article  MathSciNet  Google Scholar 

  40. ten Have G.: Structure of the nth roots of a matrix. Linear Algebra Appl. 187, 59–66 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  41. Vilenkin, N.Ya.:Combinatorics. Academic Press, New York (1971) (translated from the Russian by A. Shenitzer and S. Shenitzer)

  42. Voronin, A.F.: Partial indices of unitary and Hermitian matrix functions, Sibirsk. Mat. Zh. 51(5), 1010–1016 (2010) (in Russian) (English translation: Sib. Math. J. 51(5), 805–809, 2010)

  43. Voronin A.F.: A method for determining the partial indices of symmetric matrix functions. Sib. Math. J. 52(1), 41–53 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  44. Voronin A.F.: On a method for determining partial indices of symmetric matrix functions. Dokl. Akad. Nauk 437(4), 448–451 (2011)

    MathSciNet  Google Scholar 

  45. Yale P.B.: Automorphisms of the complex numbers. Math. Mag. 39, 135–141 (1966)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Leiba Rodman

  2. Division of Science and Mathematics, New York University Abu Dhabi, Abu Dhabi, United Arab Emirates

    Ilya M. Spitkovsky

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

Corresponding author

Correspondence to Ilya M. Spitkovsky.

Additional information

I. M. Spitkovsky was supported in part by the Summer Research Grant of the College of William and Mary and by Faculty Research funding from the Division of Science and Mathematics, New York University Abu Dhabi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rodman, L., Spitkovsky, I.M. Factorization of Matrices with Symmetries over Function Algebras. Integr. Equ. Oper. Theory 80, 469–510 (2014). https://doi.org/10.1007/s00020-014-2155-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00020-014-2155-8

Mathematics Subject Classification (2010)

Keywords

Search

Navigation