Skip to main content
Log in

The operator equationAX−XB=C with unbounded operatorsA andB and related abstract Cauchy problems

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  1. Arendt, W., Batty, C.J.K.: Tauberian theorems and stability of one-parameter semigroups. Trans. Am. Math. Soc.306, 837–852 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bellman, R.: Introduction to Matrix Analysis, New York: McGraw-Hill 1960

    MATH  Google Scholar 

  3. Belonosov, V.S.: On the indices of instability of unbounded operators. Sov. Math., Dokl.28, 541–544 (1983)

    MATH  Google Scholar 

  4. Belonosov, V.S.: Instability indices of differential operators. Math. USSR Sb.57, 507–525 (1985)

    Article  Google Scholar 

  5. Benedetto, J.J.: Spectral Synthesis, New York: Academic Press 1975

    Google Scholar 

  6. Clement, Ph. et al.: One-Parameter Semigroup. Amsterdam: North-Holland 1987

    Google Scholar 

  7. Daleckii, Ju.L.: On the asymptotic solution of a vector differential equation. Dokl. Akad. Nauk SSSR92, 881–884 (1953)

    MathSciNet  Google Scholar 

  8. Daleckij, Ju.L.: Krein, M.G.: Stability of Solutions of Differential Equations on Banach Spaces. Providence, R.I.: Am. Math. Soc. 1974

    Google Scholar 

  9. Datko, R.: Extending a theorem of A.M. Liapunov to Hilbert space. J. Math. Anal. Appl.32, 610–616 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  10. deLaubenfels, R.: Integrated semigroups,c-semigroups and the abstract Cauchy problem. Semigroup Forum41(1), 83–95 (1990)

    MATH  MathSciNet  Google Scholar 

  11. Favard, J.: Sur les équations differentielles à coefficients prespue périodique. Acta Math.51, 31–81 (1927)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hale, J.: Asymptotic Behavior of Dissipative Systems. Providence, R.I.: Am. Math. Soc. 1988

    MATH  Google Scholar 

  13. Halmos, P.: A Hilbert Space Problem Book. Heidelberg Berlin: Springer 1974

    MATH  Google Scholar 

  14. Hille, E., Phillips, R.S.: Functional Analysis and Semigroups, Providence, R.I.: Am. Math. Soc. 1957

    MATH  Google Scholar 

  15. Katznelson, Y.: An Introduction to Harmonic Analysis. New York: Dover 1976

    MATH  Google Scholar 

  16. Leeuw, K. de, Glicksberg, I.: Application of almost periodic compactification. Acta Math.105, 63–97 (1961)

    Article  MathSciNet  Google Scholar 

  17. Levitan, B.M., Zhikov, V.V.: Almost Periodic Functions and Differential Equations. Cambridge: Cambridge University Press 1982

    MATH  Google Scholar 

  18. Lumer, G., Rosenblum, M.: Linear operator equations. Proc. Am. Math. Soc.10, 32–41 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  19. Lyubich, Y.I.: Introduction to the Theory of Banach Representations of Groups. Basel: Birkhäuser 1988

    MATH  Google Scholar 

  20. Lyubich, Y.I., Vũ Quôc Phóng: Asymptotic stability of linear differential equations in Banach spaces. Stud. Math.88 37–42 (1988)

    MATH  Google Scholar 

  21. Nagel, R.: Towards a ‘Matrix Theory’ for unbounded operator matrices. Math. Z.201, 57–68 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  22. Nagel, R. (ed.): One-parameter semigroups of positive operators. (Lect. Notes Math., vol. 1184) Berlin Heidelberg New York: Springer 1986

    MATH  Google Scholar 

  23. Nambu, T.: Feedback stabilization of diffusion equations by a functional observer. J. Differ. Equations43, 257–280 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  24. Neubrander, F.: Asymptotic behavior of solutions of inhomogeneous abstract Cauchy problems. In: Lightbourne, J.H., Randin, S.M. (eds.) Proc. Conference Physical Math. and Nonlinear Partial. Diff. Equatins. Morgantown 1983. (Lect. Notes Pure Appl. Math., vol.102, pp. 157–173. New York: Marcel Dekker 1985

    Google Scholar 

  25. Neubrander, F.: Integrated semigroups and their applications to the abstract Cauchy problem Pac. J. Math.135, 111–157 (1988)

    MATH  MathSciNet  Google Scholar 

  26. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. New York: Springer 1983

    MATH  Google Scholar 

  27. Putnam, C.R.: Commutation Properties of Hilbert Space Operators and Related Topics. Berlin Heidelberg New York: Springer 1967

    Google Scholar 

  28. Rao, A.S., Hengartner, W.: On the existence of a unique almost periodic solution of an abstract differential equation. J. Lond. Math. Soc., II. Ser.8, 577–581 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  29. Rosenblum, M.: On the operator equationBX−XA=Q. Duke Math. J.23, 263–270 (1956)

    Article  MATH  MathSciNet  Google Scholar 

  30. Rosenblum, M.: The operator equationBX−XA=Q with selfadjointA andB. Proc. Am. Math. Soc.20, 115–120 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  31. Roth, W.E.: The equationAX−YB=C andAX−XB=C in matrices. Proc. Am. Math. Soc.3, 392–396 (1952)

    Article  MATH  Google Scholar 

  32. Vũ Quôc Phóng, Lyubich, Y.I.: A spectral criterion for almost periodicity of one-parameter semi-groups. (in Russian). Teor. Funks. Funkts. Anal. Prilozh.47, 36–41 (1987); English translation in J. Sov. Math.48 (No. 6), 644–648 (1990)

    Google Scholar 

  33. Zabszyk, J.: A note onC 0-semigroups, Bull. Acad. Pol. Sci., Ser. Sci. Math. Astron. Phys.23, 895–898 (1975)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Dedicated to my teacher Yu. J. Lyubich on the occasion of his 60th birthday

Rights and permissions

Reprints and permissions

About this article

Cite this article

Phóng, V.Q. The operator equationAX−XB=C with unbounded operatorsA andB and related abstract Cauchy problems. Math Z 208, 567–588 (1991). https://doi.org/10.1007/BF02571546

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02571546

Keywords

Navigation