On the exponential stability and dichotomy of C 0 -semigroups

Phóng Vũ

Studia Mathematica (1999)

  • Volume: 132, Issue: 2, page 141-149
  • ISSN: 0039-3223

Abstract

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A characterization of exponentially dichotomic and exponentially stable C 0 -semigroups in terms of solutions of an operator equation of Lyapunov type is presented. As a corollary a new and shorter proof of van Neerven’s recent characterization of exponential stability in terms of boundedness of convolutions of a semigroup with almost periodic functions is given.

How to cite

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Vũ, Phóng. "On the exponential stability and dichotomy of $C_0$-semigroups." Studia Mathematica 132.2 (1999): 141-149. <http://eudml.org/doc/216591>.

@article{Vũ1999,
abstract = {A characterization of exponentially dichotomic and exponentially stable $C_0$-semigroups in terms of solutions of an operator equation of Lyapunov type is presented. As a corollary a new and shorter proof of van Neerven’s recent characterization of exponential stability in terms of boundedness of convolutions of a semigroup with almost periodic functions is given.},
author = {Vũ, Phóng},
journal = {Studia Mathematica},
keywords = {exponentially dichotomic and exponentially stable -semigroups; operator equation of Lyapunov type; exponential stability; boundedness of convolutions of a semigroup with almost periodic functions},
language = {eng},
number = {2},
pages = {141-149},
title = {On the exponential stability and dichotomy of $C_0$-semigroups},
url = {http://eudml.org/doc/216591},
volume = {132},
year = {1999},
}

TY - JOUR
AU - Vũ, Phóng
TI - On the exponential stability and dichotomy of $C_0$-semigroups
JO - Studia Mathematica
PY - 1999
VL - 132
IS - 2
SP - 141
EP - 149
AB - A characterization of exponentially dichotomic and exponentially stable $C_0$-semigroups in terms of solutions of an operator equation of Lyapunov type is presented. As a corollary a new and shorter proof of van Neerven’s recent characterization of exponential stability in terms of boundedness of convolutions of a semigroup with almost periodic functions is given.
LA - eng
KW - exponentially dichotomic and exponentially stable -semigroups; operator equation of Lyapunov type; exponential stability; boundedness of convolutions of a semigroup with almost periodic functions
UR - http://eudml.org/doc/216591
ER -

References

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  8. [8] S. C. Lin and S. Y. Shaw, On the operator equations Ax=q and SX-XT=Q, ibid. 77 (1988), 352-363. Zbl0647.47028
  9. [9] G. Lumer and M. Rosenblum, Linear operator equations, Proc. Amer. Math. Soc. 10 (1959), 32-41. Zbl0133.07903
  10. [10] R. Nagel (ed.), One-Parameter Semigroups of Positive Operators, Lecture Notes in Math. 1184, Springer, Berlin, 1986. Zbl0585.47030
  11. [11] J. M. A. M. van Neerven, The Asymptotic Behavior of a Semigroup of Linear Operators, Birkhäuser, Basel, 1996. Zbl0905.47001
  12. [12] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983. 
  13. [13] J. Prüss, On the spectrum of C 0 -semigroups, Trans. Amer. Math. Soc. 284 (1984), 847-857. Zbl0572.47030
  14. [14] C. R. Putnam, Commutation Properties of Hilbert Space Operators and Related Topics, Springer, New York, 1967. Zbl0149.35104
  15. [15] M. Rosenblum, On the operator equation BX-XA=Q, Duke Math. J. 23 (1956), 263-269. Zbl0073.33003
  16. [16] Vũ Quôc Phóng, The operator equation AX-XB=C with unbounded operators A and B and related abstract Cauchy problems, Math. Z. 208 (1991), 567-588. 
  17. [17] Vũ Quôc Phóng, On the spectrum, complete trajectories, and asymptotic stability of linear semidynamical systems, J. Differential Equations 105 (1993), 30-45. 

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