Skip to main content
SpringerLink
Log in

Inner Outer Factorization of Wide Rational Matrix Valued Functions on the Half Plane

  • Conference paper
  • First Online:
Operator Theory, Functional Analysis and Applications

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 282))

  • 745 Accesses

Abstract

The main purpose of this note is to use operator methods to solve a rational inner-outer factorization problem for wide functions. It is believed that this will provide valuable insight into the inner-outer factorization problem. Our approach involves Wiener–Hopf operators, Hankel operators and invariant subspaces for the backward shift. It should be emphasized that the formulas for the inner and outer factor are derived in a computational manner.

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 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
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

References

  1. D. Alpay, I. Gohberg, Unitary rational matrix functions, in Topics in Interpolation Theory of Rational Matrix-valued Functions. Operator Theory: Advances and Applications, vol. 33 (Birkhäuser Verlag, Basel, 1988), pp. 175–222

    Google Scholar 

  2. D.Z. Arov, H. Dym, J-Contractive Matrix Valued Functions and Related Topics (Cambridge University Press, Cambridge, 2008)

    Book  Google Scholar 

  3. H. Bart, I. Gohberg, M.A. Kaashoek, A.C.M. Ran, A State Space Approach to Canonical Factorization: Convolution Equations and Mathematical Systems. Operator Theory: Advances and Applications, vol. 200 (Birkhäuser Verlag, Basel, 2010)

    Google Scholar 

  4. T. Chen, B.A. Francis, Spectral and inner-outer-factorizations of rational matrices. SIAM J. Matrix Anal. Appl. 10, 1–17 (1989)

    Article  MathSciNet  Google Scholar 

  5. H. Dym, Linear Algebra in Action. Graduate Studies in Mathematics, vol. 78 (American Mathematical Society, Providence, 2007)

    Google Scholar 

  6. A.E. Frazho, W. Bosri, An Operator Perspective on Signals and Systems. Operator Theory: Advances and Applications, vol. 204 (Birkhäuser Verlag, Basel, 2010)

    Google Scholar 

  7. C. Foias, A. Frazho, The Commutant Lifting Approach to Interpolation Problems. Operator Theory: Advances and Applications, vol. 44 (Birkhäuser Verlag, Basel, 1990)

    Google Scholar 

  8. C. Foias, A. Frazho, I. Gohberg, M.A. Kaashoek, Metric Constrained Interpolation, Commutant Lifting and Systems. Operator Theory: Advances and Applications, vol. 100 (Birkhäuser Verlag, Basel, 1998)

    Google Scholar 

  9. A.E. Frazho, A.C.M. Ran, A note on inner-outer factorization for wide matrix-valued functions, in: Operator Theory, Analysis and the State Space Approach. Operator Theory: Advances and Applications, vol. 271 (Birkhäuser Verlag, Basel, 2018), pp. 201–214

    Google Scholar 

  10. A.E. Frazho, M.A. Kaashoek, A.C.M. Ran, Rational matrix solutions of a Bezout type equation on the half plane, in: Advances in Structured Operator Theory and Related Areas. Operator Theory: Advances and Applications, vol. 237 (Birkhäuser Verlag, Basel, 2013), pp. 145–160

    Google Scholar 

  11. I. Gohberg, S. Goldberg, M.A. Kaashoek, Classes of Linear Operators, Volume I. Operator Theory: Advances and Applications, vol. 49 (Birkhäuser Verlag, Basel, 1990)

    Google Scholar 

  12. I. Gohberg, S. Goldberg, M.A. Kaashoek, Classes of Linear Operators, Volume II. Operator Theory: Advances and Applications, vol. 63 (Birkhäuser Verlag, Basel, 1993)

    Google Scholar 

  13. V. Ionescu, C. Oară, M. Weiss, Generalized Riccati Theory and Robust Control, A Popov Function Approach (Wiley, Chichester, 1999)

    MATH  Google Scholar 

  14. P.D. Lax, Translation invariant spaces. Acta Math. 101, 163–178 (1959)

    Article  MathSciNet  Google Scholar 

  15. N.K. Nikol’skii, Treatise on the Shift Operator. Grundlehren, vol. 273 (Springer, Berlin, 1986)

    Google Scholar 

  16. C. Oară, A. Varga, Computation of general inner–outer and spectral factorizations. IEEE Trans. Autom. Control 45, 2307–2325 (2000)

    Article  MathSciNet  Google Scholar 

  17. T. Reis, M. Voigt, Inner-outer factorization for differential-algebraic systems. Math. Control Signals Syst. 30, Art. 15, 19pp. (2018)

    Google Scholar 

  18. M. Rosenblum, J. Rovnyak, Hardy Classes and Operator Theory (Oxford University Press, Oxford, 1985)

    MATH  Google Scholar 

  19. B. Sz.-Nagy, C. Foias, Harmonic Analysis of Operators on Hilbert Space (North-Holland, Amsterdam, 1970)

    Google Scholar 

  20. B. Sz.-Nagy, C. Foias, H. Bercovici, L. Kérchy, Harmonic Analysis of Operators on Hilbert Space (Springer, New York, 2010)

    Google Scholar 

  21. F.-B. Yeh, L.-F. Wei, Inner-outer factorizations of right-invertible real-rational matrices. Syst. Control Lett. 14, 31–36 (1990)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    A. E. Frazho

  2. Department of Mathematics, Faculty of Science, Vrije Universiteit Amsterdam, Amsterdam, The Netherlands

    A. C. M. Ran

  3. Research Focus: Pure and Applied Analytics, 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. A. C. M. Ran
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. C. M. Ran .

Editor information

Editors and Affiliations

  1. Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal

    M. Amélia Bastos

  2. Departamento de Matemática, Universidade de Aveiro, Aveiro, Portugal

    Luís Castro

  3. Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Lisboa, Portugal

    Alexei Yu. Karlovich

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Frazho, A.E., Ran, A.C.M. (2021). Inner Outer Factorization of Wide Rational Matrix Valued Functions on the Half Plane. In: Bastos, M.A., Castro, L., Karlovich, A.Y. (eds) Operator Theory, Functional Analysis and Applications. Operator Theory: Advances and Applications, vol 282. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-51945-2_11

Download citation

Publish with us

Policies and ethics

Search

Navigation