Abstract
Let T t : X → X be a C 0-semigroup with generator A. We prove that if the abscissa of uniform boundedness of the resolvent s 0(A) is greater than zero then for each nondecreasing function h(s): ℝ+ → R + there are x′ ∈ X′ and x ∈ X satisfying ∫ ∞0 h(|〈x′, T x x〉|)dt = ∞. If iℝ ∩ Sp(A) ≠ Ø then such x may be taken in D(A ∞).
Article PDF
Similar content being viewed by others
References
Muller V., “Local spectral radius formula for operators in Banach spaces,” Czechoslovak Math. J., 38, No. 4, 726–729 (1988).
van Neerven J. M. A. M., “On the orbits of an operator with spectral radius one,” Czechoslovak Math. J., 45, No. 3, 495–502 (1995).
Datko R., “Extending a theorem of A. M. Lyapunov to Hilbert space,” J. Math. Anal. Appl., 32, No. 3, 610–616 (1970).
Pazy A., “On the applicability of Lyapunov’s theorem in Hilbert space,” SIAM J. Math. Anal., 3, No. 2, 291–294 (1972).
Rolewicz S., “On uniform N-equistability,” J. Math. Anal. Appl., 115, No. 2, 434–441 (1986).
Storozhuk K. V., “On the Rolewicz theorem for evolution operators,” Proc. Amer. Math. Soc., 135, No. 6, 1861–1863 (2007).
van Neerven J. M. A. M., The Asymptotic Behaviour of Semigroups of Linear Operators, Birkhäuser, Basel (1996).
Pritchard A. J. and Zabczyk J., “Stability and stabilizability of infinite-dimensional systems,” SIAM Rev., 23, No. 1, 25–52 (1981).
Huang Fa Lun, “Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces,” Ann. Differential Equations, 1, No. 1, 43–56 (1985).
Weiss G., “Weak Lp-stability of a linear semigroup on a Hilbert space implies exponential stability,” J. Differential Equations, 76, No. 2, 269–285 (1988).
Greiner G., Voigt J., and Wolff M., “On the spectral bound of the generator of semigroups of positive operators,” J. Operator Theory, 5, No. 2, 245–256 (1981).
van Neerven J. M. A. M., Straub B., and Weis L., “On the asymptotic behaviour of a semigroup of linear operators,” Indag. Math. (N. S.), 6, No. 4, 435–476 (1995).
Gearhart L., “Spectral theory for contraction semigroups on Hilbert spaces,” Trans. Amer. Math. Soc., 236, 385–394 (1978).
Arendt W., Graboch A., Greiner G. et al., One-Parameter Semigroups of Positive Operators, Springer-Verlag, Berlin (1986) (Lecture Notes in Math.; 1184).
Neubrander F., “Laplace transform and asymptotic behavior of strongly continuous semigroups,” Houston J. Math., 12, No. 4, 549–561 (1986).
Foiaş C., “Sur une question de M. Reghis,” An. Univ. Timişoara Ser. ŞSi. Mat., 9, No. 2, 111–114 (1973).
Zabczyk J., “A note on C0-semigroups,” Bull. Acad. Polon. Sci., 23, No. 8, 895–898 (1975).
Wrobel V., “Asymptotic behavior of C0-semigroups in B-convex spaces,” Indiana Univ. Math. J., 38, No. 1, 101–114 (1989).
Batkai A., Engel K.-J., Pruss J., and Schnaubelt R., “Polynomial stability of operator semigroups,” Math. Nachr., 279, No. 13–14, 1425–1440 (2006).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text Copyright © 2010 Storozhuk K. V.
__________
Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 51, No. 2, pp. 410–419, March–April, 2010.
Rights and permissions
About this article
Cite this article
Storozhuk, K.V. Obstructions to the uniform stability of a C 0-semigroup. Sib Math J 51, 330–337 (2010). https://doi.org/10.1007/s11202-010-0034-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11202-010-0034-3