Skip to main content

Inverting Structured Operators Related to Toeplitz Plus Hankel Operators

  • Chapter
  • First Online:
  • 519 Accesses

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

Abstract

In this paper the Ellis–Gohberg–Lay theorem on inversion of certain Toeplitz plus Hankel operators is derived as a corollary of an abstract inversion theorem for a certain class of structured operators. The main results also cover the inversion theorems considered in [6].

Mathematics Subject Classification (2010). Primary 47A62, 47B35; Secondary 47A50, 15A09, 65F05

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

Buying options

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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

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

    Google Scholar 

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

    Google Scholar 

  3. M.J. Corless and A.E. Frazho, Linear systems and control, Marcel Dekker, Inc., New York, 2003.

    Google Scholar 

  4. R.L. Ellis and I. Gohberg, Orthogonal systems and convolution operators, OT 140, Birkhäuser Verlag, Basel, 2003.

    Google Scholar 

  5. R.L. Ellis, I. Gohberg, and D.C. Lay, Infinite analogues of block Toeplitz matrices and related orthogonal functions, Integral Equations and Operator Theory 22 (1995), 375–419.

    Article  MathSciNet  MATH  Google Scholar 

  6. A.E. Frazho and M.A. Kaashoek, A contractive operator view on an inversion formula of Gohberg–Heinig, in: Topics in Operator Theory I. Operators, matrices and analytic functions, OT 202, Birkhäuser Verlag, Basel, 2010, pp. 223–252.

    Google Scholar 

  7. I. Gohberg, S. Goldberg, and M.A. Kaashoek, Classes of Linear Operators, Volume II, OT 63, Birkhäuser Verlag, Basel, 1993.

    Google Scholar 

  8. I. Gohberg, G. Heinig, The inversion of finite Toeplitz matrices consisting of elements of a non-commutative algebra, Rev. Roum. Math. Pures et Appl. 20 (1974), 623– 663 (in Russian); English transl. in: Convolution Equations and Singular Integral Operators, (eds. L. Lerer, V. Olshevsky, I.M. Spitkovsky), OT 206, Birkhäuser Verlag, Basel, 2010, pp. 7–46.

    Google Scholar 

  9. I.C. Gohberg and M.G. Krein, Systems of integral equations with kernels depending on the difference of arguments, Uspekhi Math. Nauk 13 2(80) (1958), 3–72 (Russian); English Transl., Amer. Math. Soc. Transl. (Series 2) 14 (1960), 217–287.

    Google Scholar 

  10. G.J. Groenewald and M.A. Kaashoek, A Gohberg–Heinig type inversion formula involving Hankel operators,in: Interpolation, Schur functions and moment problems, OT 165, Birkhäuser Verlag, Basel, 2005, pp. 291–302.

    Google Scholar 

  11. G. Heinig and K. Rost, Algebraic methods for Toeplitz-like matrices and operators, Akademie-Verlag, Berlin, 1984.

    Google Scholar 

  12. T. Kailath and A.H. Sayed, Displacement structure: Theory and applications, SIAM Rev. 37 (1995), 297–386.

    Article  MathSciNet  MATH  Google Scholar 

  13. T. Kailath and A.H. Sayed (editors), Fast Reliable Algorithms for Matrices with Structure, SIAM, Philadelphia, 1999.

    Google Scholar 

  14. I. Koltracht, B.A. Kon, and L. Lerer, Inversion of structured operators, Integral equations and Operator Theory 20 (1994), 410–448.

    Article  MathSciNet  MATH  Google Scholar 

  15. V. Olshevsky (editor), Structured matrices in mathematics, Computer Science, and Engineering, Contempary Math. Series 280, 281, Amer. Math. Soc. 2001.

    Google Scholar 

  16. V.Y. Pan, Structured matrices and polynomials, Birkhäuser Boston, 2001.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to M. A. Kaashoek .

Editor information

Editors and Affiliations

Additional information

Dedicated to our dear friend Leonia Lerer, on the occasion of his 70th birthday.

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Basel

About this chapter

Cite this chapter

Kaashoek, M.A., van Schagen, F. (2013). Inverting Structured Operators Related to Toeplitz Plus Hankel Operators. 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_12

Download citation

Publish with us

Policies and ethics