Skip to main content
SpringerLink
Log in

Normal Matrices in Degenerate Indefinite Inner Product Spaces

  • Conference paper
Operator Theory in Inner Product Spaces

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 175))

Abstract

Complex matrices that are structured with respect to a possibly degenerate indefinite inner product are studied. Based on the theory of linear relations, the notion of an adjoint is introduced: the adjoint of a matrix is defined as a linear relation which is a matrix if and only if the inner product is nondegenerate. This notion is then used to give alternative definitions of selfadjoint and unitary matrices in degenerate inner product spaces and it is shown that those coincide with the definitions that have been used in the literature before. Finally, a new definition for normal matrices is given which allows the generalization of an extension result for positive invariant subspaces from the case of nondegenerate inner products to the case of degenerate inner products.

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
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. G. Ammar, C. Mehl, and V. Mehrmann, Schur-like Forms for Matrix Lie Groups, Lie Algebras and Jordan Algebras. Linear Algebra Appl. 287 (1999), 11–39.

    Article  MATH  MathSciNet  Google Scholar 

  2. T.Ya. Azizov, Completely Continuous Operators that are Selfadjoint with Respect to a Degenerate Indefinite Metric (Russian). Mat. Issled. 7 (1972), 237–240, 259.

    MATH  MathSciNet  Google Scholar 

  3. T.Ya. Azizov and I.S. Iohvidov, Linear Operators in Spaces with an Indefinite Metric. John Wiley and Sons, Ltd., Chichester, 1989. (Translated from Russian.)

    Google Scholar 

  4. V. Bolotnikov, C.K. Li, P. Meade, C. Mehl, and L. Rodman, Shells of Matrices in Indefinite Inner Product Spaces. Electron. J. Linear Algebra 9 (2002), 67–92.

    MATH  MathSciNet  Google Scholar 

  5. R. Cross, Multivalued Linear Operators. Marcel Dekker Inc., 1998.

    Google Scholar 

  6. A. Dijksma and H.S.V. de Snoo, Symmetric and Selfadjoint Relations in Krein Spaces I. Oper. Theory Adv. Appl. 24 (1987), 145–166.

    Google Scholar 

  7. A. Dijksma and H.S.V. de Snoo, Symmetric and Selfadjoint Relations in Krein Spaces II. Ann. Acad. Sci. Fenn. Math. 12 (1987), 199–216.

    MATH  Google Scholar 

  8. I. Gohberg, P. Lancaster, and L. Rodman, Matrices and Indefinite Scalar Products. Birkhäuser, 1983.

    Google Scholar 

  9. I. Gohberg, P. Lancaster, and L. Rodman, Indefinite Linear Algebra. Birkhäuser, 2005.

    Google Scholar 

  10. I. Gohberg and B. Reichstein, On Classification of Normal Matrices in an Indefinite Scalar Product. Integral Equations Operator Theory 13 (1990), 364–394.

    Article  MATH  MathSciNet  Google Scholar 

  11. M. Kaltenbäck and H. Woracek, Selfadjoint Extensions of Symmetric operators in Degenerated Inner Product Spaces, Integral Equations Operator Theory 28 (1997), 289–320.

    Article  MATH  MathSciNet  Google Scholar 

  12. P. Lancaster, A.S. Markus, and P. Zizler, The Order of Neutrality for Linear Operators on Inner Product Spaces. Linear Algebra Appl. 259 (1997), 25–29.

    Article  MATH  MathSciNet  Google Scholar 

  13. P. Lancaster and L. Rodman, Algebraic Riccati Equations. Clarendon Press, 1995.

    Google Scholar 

  14. H. Langer, R. Mennicken, and C. Tretter, A Self-Adjoint Linear Pencil Q — λP of Ordinary Differential Operators. Methods Funct. Anal. Topology 2 (1996), 38–54.

    MATH  MathSciNet  Google Scholar 

  15. H. Langer, Invariante Teilräume definisierbarer J-selbstadjungierter Operatoren. Ann. Acad. Sci. Fenn. Ser. A.I. Math. 475 (1971), 1–23

    Google Scholar 

  16. C.K. Li, N.K. Tsing, and F. Uhlig, Numerical Ranges of an Operator on an Indefinite Inner Product Space. Electron. J. Linear Algebra 1 (1996), 1–17.

    MathSciNet  Google Scholar 

  17. C. Mehl, A. Ran, and L. Rodman, Semidefinite Invariant Subspaces: Degenerate Inner Products. Oper. Theory Adv. Appl. 149 (2004), 467–486.

    MathSciNet  Google Scholar 

  18. C. Mehl, A. Ran, and L. Rodman, Hyponormal Matrices and Semidefinite Invariant Subspaces in Indefinite Inner Products. Electron. J. Linear Algebra 11 (2004), 192–204.

    MATH  MathSciNet  Google Scholar 

  19. C. Mehl, A. Ran, and L. Rodman, Extension to Maximal Semidefinite Invariant Subspaces for Hyponormal Matrices in Indefinite Inner Products. Linear Algebra Appl. 421 (2007), 110–116.

    Article  MATH  MathSciNet  Google Scholar 

  20. C. Mehl and L. Rodman, Symmetric Matrices with Respect to Sesquilinear Forms. Linear Algebra Appl. 349 (2002), 55–75.

    Article  MATH  MathSciNet  Google Scholar 

  21. V. Mehrmann, Existence, Uniqueness, and Stability of Solutions to Singular Linear Quadratic Optimal Control Problems. Linear Algebra Appl. 121 (1989), 291–331.

    Article  MATH  MathSciNet  Google Scholar 

  22. B. C. Ritsner, The Theory of Linear Relations. (Russian) Voronezh, Dep. VINITI, No. 846-82, 1982.

    Google Scholar 

  23. H. Woracek, Resolvent Matrices in Degenerated Inner Product Spaces. Math. Nachr. 213 (2000), 155–175.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, MA 4-5, Technische Universität Berlin, Str. des 17. Juni 136, D-10623, Berlin, Germany

    Christian Mehl

  2. Institut für Mathematik, MA 6-3, Technische Universität Berlin, Str. des 17. Juni 136, D-10623, Berlin, Germany

    Carsten Trunk

Authors
  1. Christian Mehl
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Carsten Trunk
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Mathematik, MA 6-4, Technische Universität Berlin, Straße des 17. Juni 136, D-10623, Berlin

    Karl-Heinz Förster , Peter Jonas  & Carsten Trunk ,  & 

  2. Mathematik, Technische Universität Wien, Wiedner Hauptstrasse 8-10/1411, A-1040, Wien

    Heinz Langer

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Birkhäuser Verlag Basel/Switzerland

About this paper

Cite this paper

Mehl, C., Trunk, C. (2007). Normal Matrices in Degenerate Indefinite Inner Product Spaces. In: Förster, KH., Jonas, P., Langer, H., Trunk, C. (eds) Operator Theory in Inner Product Spaces. Operator Theory: Advances and Applications, vol 175. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8270-4_11

Download citation

Publish with us

Policies and ethics

Search

Navigation