Skip to main content
SpringerLink
Log in

On the adjoint of operator matrices with unbounded entries II

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

In this paper, the adjoint of a densely defined block operator matrix

$\mathcal{L} = [_{CD}^{AB} ]$

in a Hilbert space X × X is studied and the sufficient conditions under which the equality

$\mathcal{L}* = [_{B*D*}^{A*C*} ]$

holds are obtained through applying Frobenius-Schur factorization.

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. Atkinson, F. V., Langer, H., Mennicken, R., et al.: The essential spectrum of some matrix operators. Math. Nachr., 167, 5–20 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  2. Chen, A., Jin, G. H.: On the (α, β)-essential spectrum of upper triangular operator matrices. Acta Math. Sin., Chin. Series, 57, 569–580 (2014)

    Article  MATH  Google Scholar 

  3. Hardt, V., Konstantinov, A., Mennicken, R.: On the spectrum of the product of closed operators. Math. Nachr., 215, 91–102 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Kato, T.: Perturbation Theory for Linear Operators (Second Corrected Printing of Second Edition), Springer-Verlag, New York, 1984

    Google Scholar 

  5. Langer, H., Ran, A. C. M., Rotten, B. A.: Invariant subspaces of infinite dimensional Hamiltonians and solutions of the corresponding Riccati equations. Oper. Theory Adv. Appl., 130, 230–254 (2001)

    Google Scholar 

  6. Reed, M., Simon, B.: Methods of Modern Mathematical Physics I (Functional Analysis), Academic Press, New York, 1978

    Google Scholar 

  7. Shen, J. L., Chen, A.: The spectral properties of quasi-*-A(n) operators. Acta Math. Sin., Chin. Series, 56, 537–544 (2013)

    MATH  Google Scholar 

  8. Tretter, C.: Spectral inclusion for unbounded block operator matrices. J. Funct. Anal., 256, 3806–3829 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. Tretter, C.: Spectral Theory of Block Operator Matrices and Applications, Imperial College Press, London, 2008

    Book  MATH  Google Scholar 

  10. Weidmann, J.: Linear Operators in Hilbert Spaces, Springer-Verlag, New York, 1980

    Book  MATH  Google Scholar 

  11. Wu, D. Y., Chen, A.: Invertibility of nonnegative Hamiltonian operator with unbounded entries. J. Math. Anal. Appl., 373, 410–413 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  12. Wu, D. Y., Chen, A.: Spectral Theory of Block Operator Matrices and Application (in Chinese), Science Press, Beijing, 2013

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Inner Mongolia University, Hohhot, 010021, P. R. China

    De Yu Wu

  2. Department of Mathematics, Hohhot University for Nationalities, Hohhot, 010050, P. R. China

    Alatancang Chen

Authors
  1. De Yu Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alatancang Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to De Yu Wu.

Additional information

Supported by NSFC (Grant Nos. 11101200, 11371185, 2013ZD01)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, D.Y., Chen, A. On the adjoint of operator matrices with unbounded entries II. Acta. Math. Sin.-English Ser. 31, 995–1002 (2015). https://doi.org/10.1007/s10114-015-4275-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-015-4275-8

Keywords

MR(2010) Subject Classification

Search

Navigation