On the Roughness of Quasinilpotency Property of One-parameter Semigroups

Preda, Ciprian

Canadian mathematical bulletin, Tome 60 (2017) no. 2, pp. 364-371

Résumé

Let $\mathbf{S}:={{\{S(t)\}}_{t\ge 0}}$ be a ${{\text{C}}_{0}}$ -semigroup of quasinilpotent operators (i.e., $\sigma (S(t))=\{0\}$ for each $t>0$ ). In dynamical systems theory the above quasinilpotency property is equivalent to a very strong concept of stability for the solutions of autonomous systems. This concept is frequently called superstability and weakens the classical finite time extinction property (roughly speaking, disappearing solutions). We show that under some assumptions, the quasinilpotency, or equivalently, the superstability property of a ${{\text{C}}_{0}}$ -semigroup is preserved under the perturbations of its infinitesimal generator.

Détail

DOI : 10.4153/CMB-2016-088-0

Mots-clés : 34D05, 34D10, 34E10, one-parameter semigroups, quasinilpotency, superstability, essential spectrum.

Bibliographie

[1] [1] Balakrishnan, A. V., Superstability of systems. Appl. Math. Comput. 164(2005), no. 2, 321–326. Google Scholar

[2] [2] Blake, M. D., A spectral bound for asymptotically norm-continuous semigroups. J. Operator Theory 45(2001), no. 1, 111–130. Google Scholar

[3] [3] Brendle, S., On the asymptotic behavior of perturbed strongly continuous semigroups. Math. Nachr. 226(2001), 35–47. http://dx.doi.org/10.1002/1522-2616(200106)226:1<35::AID-MANA35>3.0.CO;2-R 3.0.CO;2-R>Google Scholar

[4] [4] Brendle, S., Nagel, R., and Poland, J., On the spectral mapping theorem for perturbed strongly continuous semigroups. Arch. Math. (Basel) 74(2000), no. 5, 365–378. Google Scholar | DOI

[5] [5] Engel, K. and Nagel, R., One-parameter semigroups for linear evolution equations. Graduate Texts in Mathematics, 194. Springer-Verlag, New York, 2000. Google Scholar

[6] [6] Hille, E. and Phillips, R. S., Functional analysis and semi-groups. American Mathematical Society Colloquium Publications, 31, American Mathematical Society, Providence, RI, 1957. Google Scholar

[7] [7] Hislop, P. D. and Sigal, I. M., Introduction to spectral theory: with applications to Schrb'dinger operators. Applied Mathematical Sciences, 113, Springer-Verlag, New York, 1996. Google Scholar | DOI

[8] [8] Nagel, R. and Poland, J., The critical spectrum of a strongly continuous semigroup. Adv. Math. 152(2000), 120-133. Google Scholar | DOI

[9] [9] Nagel, R. and Râbiger, F., Superstable operators on Banach spaces. Israel J. Math. 81(1993), 213–226. Google Scholar | DOI

[10] [10] Phillips, R. S., Spectral theory for semigroups of linear operators. Trans. Amer. Math. Soc. 71(1951), 393–415. Google Scholar | DOI

[11] [11] Phillips, R. S., Perturbation theory for semigroups of linear operators. Trans. Amer.Math. Soc. 74(1953), 199–221. Google Scholar | DOI

[12] [12] Râbiger, F. and Wolff, M., Superstable semigroups of operators. Indag. Math. 6(1995), 481–494. Google Scholar | DOI

[13] [13] Voigt, J., A perturbation theorem for the essential spectral radius of strongly continuous semigroups. Monatsh. Math. 90(1980), 153–161. Google Scholar | DOI

[14] [14] Voigt, J., Stability of the essential type of strongly continuous semigroups. Trans. Steklov Inst. Math. 203(1994), 469–477. Google Scholar