Skip to main content
SpringerLink
Log in

Rational Matrix Solutions of a Bezout Type Equation on the Half-plane

  • Chapter
  • First Online:
Advances in Structured Operator Theory and Related Areas

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 237))

Abstract

A state space description is given of all stable rational matrix solutions of a general rational Bezout type equation on the right half-plane. Included are a state space formula for a particular solution satisfying a certain H 2 minimality condition, a state space formula for the inner function describing the null space of the multiplication operator corresponding to the Bezout equation, and a parameterization of all solutions using the particular solution and this inner function. A state space version of the related Tolokonnikov lemma is also presented.

Mathematics Subject Classification (2010). Primary 47B35, 39B42; Secondary 47A68, 93B28

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 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
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. M. Bakonyi and H.J. Woerdeman, Matrix completions, moments, and sums of Hermitian squares, Princeton University Press, 2010.

    Google Scholar 

  2. H. Bart, I. Gohberg, M.A. Kaashoek, and A.C.M. Ran, Factorization of matrix and operator functions: the state space method, OT 178, Birkhäuser Verlag, Basel, 2008.

    Google Scholar 

  3. H. Bart, I. Gohberg, M.A. Kaashoek, and A.C.M. Ran, A state space approach to canonical factorization: convolution equations and mathematical systems, OT 200, Birkhäuser Verlag, Basel, 2010.

    Google Scholar 

  4. L. Carlson, Interpolation by bounded analytic functions and the corona problem, Ann. Math. 76 (1962), 547–559.

    Article  Google Scholar 

  5. A.E. Frazho, M.A. Kaashoek, and A.C.M. Ran, The non-symmetric discrete algebraic Riccati equation and canonical factorization of rational matrix functions on the unit circle, Integral Equations and Operator Theory 66 (2010), 215–229.

    Article  MathSciNet  MATH  Google Scholar 

  6. A.E. Frazho, M.A. Kaashoek and A.C.M. Ran, Right invertible multiplication operators and stable rational matrix solutions to an associate Bezout equation, I. the least squares solution. Integral Equations and Operator Theory 70 (2011), 395–418.

    Google Scholar 

  7. C. Foias, A. Frazho, I. Gohberg, and M.A. Kaashoek, Metric constrained interpolation, commutant lifting and systems, Birkhäuser Verlag, Basel, 1998;

    Google Scholar 

  8. A.E. Frazho, M.A. Kaashoek and A.C.M. Ran, Right invertible multiplication operators and stable rational matrix solutions to an associate Bezout equation, II: Description of all solutions, Operators and Matrices 6 (2012), 833–857.

    Google Scholar 

  9. P. Fuhrmann, On the corona theorem and its applications to spectral problems in Hilbert space, Trans. Amer. Math. Soc. 132 (1968), 55–66.

    Article  MathSciNet  MATH  Google Scholar 

  10. P.A. Fuhrmann and R. Ober, On coprime factorizations, in: Operator Theory and its Applications. The Tsuyoshi Ando Anniversary Volume. OT 62, Birkhäuser Verlag, Basel, 1993, pp. 39–75.

    Google Scholar 

  11. T. Georgiou and M.C. Smith, Optimal robustness in the gap metric, IEEE Trans. Automatic Control 35 (1990), 673–686.

    Article  MathSciNet  MATH  Google Scholar 

  12. I. Gohberg, S. Goldberg, and M.A. Kaashoek, Classes of Linear Operators, Volume I, OT 49, Birkhäuser Verlag, Basel, 1990.

    Google Scholar 

  13. G. Gu and E.F. Badran, Optimal design for channel equalization via the filterbank approach, IEEE Trans.Signal Proc. 52 (2004), 536–545.

    Article  MathSciNet  Google Scholar 

  14. G. Gu and L. Li, Worst-case design for optimal channel equalization in filterbank transceivers, IEEE Trans. Signal Proc. 51 (2003), 2424–2435.

    Article  Google Scholar 

  15. J.W. Helton, Operator Theory, analytic functions, matrices and electrical engineering, Regional Conference Series in Mathematics 68, Amer. Math. Soc. Providence RI, 1987.

    Google Scholar 

  16. N.K. Nikol’skii, Treatise on the shift operator, Grundlehren 273, Springer Verlag, Berlin 1986.

    Google Scholar 

  17. V.V. Peller, Hankel Operators and their Applications, Springer Monographs in Mathematics, Springer 2003.

    Google Scholar 

  18. V.A. Tolokonnikov, Estimates in Carlson’s corona theorem. Ideals of the algebra \( H^{\infty} \) , the problem of Szekefalvi-Nagy, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 113 (1981), 178–1981 (Russian).

    Google Scholar 

  19. S. Treil, Lower bounds in the matrix corona theorem and the codimension one conjecture, GAFA 14 (2004), 1118–1133.

    Article  MathSciNet  MATH  Google Scholar 

  20. S. Treil and B.D. Wick, The matrix-valued \( H^{p} \) corona problem in the disk and polydisk, J. Funct. Anal. 226 (2005), 138–172.

    Article  MathSciNet  MATH  Google Scholar 

  21. T.T. Trent and X. Zhang, A matricial corona theorem, Proc. Amer. Math. Soc. 134 (2006), 2549–2558.

    Article  MathSciNet  MATH  Google Scholar 

  22. T.T. Trent, An algorithm for the corona solutions on \( H^{\infty}(D) \), Integral Equations and Operator Theory 59 (2007), 421–435.

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Vidyasagar, Control system synthesis: a factorization approach, The MIT Press, Cambridge, MA, 1985.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Aeronautics and Astronautics, Purdue University, West Lafayette, IN, 47907, USA

    A. E. Frazho

  2. Department of Mathematics Faculty of Sciences, VU University, De Boelelaan 1081a, NL-1081 HV, Amsterdam, The Netherlands

    M. A. Kaashoek & A. C. M. Ran

  3. Unit for BMI, North-West University, Potchefstroom, South Africa

    A. C. M. Ran

Authors
  1. A. E. Frazho
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. A. Kaashoek
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. C. M. Ran
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. E. Frazho .

Editor information

Editors and Affiliations

  1. Computer Sciences, Free University Amsterdam, Amsterdam, Noord-Holland, The Netherlands

    Marinus A. Kaashoek

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

    Leiba Rodman

  3. Mathematics Department, Drexel University, Philadelphia, Pennsylvania, USA

    Hugo J. Woerdeman

Additional information

Dedicated to Leonia Lerer on the occasion of his 70th birthday, in friendship

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Basel

About this chapter

Cite this chapter

Frazho, A.E., Kaashoek, M.A., Ran, A.C.M. (2013). Rational Matrix Solutions of a Bezout Type Equation on the Half-plane. In: Kaashoek, M., Rodman, L., Woerdeman, H. (eds) Advances in Structured Operator Theory and Related Areas. Operator Theory: Advances and Applications, vol 237. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0639-8_11

Download citation

Publish with us

Policies and ethics

Search

Navigation