Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-23T23:05:42.284Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Roughness of Quasinilpotency Property of One-parameter Semigroups

Published online by Cambridge University Press:  20 November 2018

Ciprian Preda
Ciprian Preda*
Affiliation:
West University of Timişoara, Bd. V. Pârvan, No. 4, Timişoara 300223, Romania. e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

34D0534D1034E10one-parameter semigroupsquasinilpotency, superstability, essential spectrum.
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 60 , Issue 2 , 01 June 2017 , pp. 364 - 371
Copyright
Copyright © Canadian Mathematical Society 2017

References

[1] Balakrishnan, A. V., Superstability of systems. Appl. Math. Comput. 164(2005), no. 2, 321–326. Google Scholar
[2] Blake, M. D., A spectral bound for asymptotically norm-continuous semigroups. J. Operator Theory 45(2001), no. 1, 111–130. Google Scholar
[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] 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. http://dx.doi.org/10.1007/s000130050456 Google Scholar
[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] 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] 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. http://dx.doi.org/10.1007/978-1-4612-0741-2 Google Scholar
[8] Nagel, R. and Poland, J., The critical spectrum of a strongly continuous semigroup. Adv. Math. 152(2000), 120-133. http://dx.doi.org/10.1006/aima.1998.1 893 Google Scholar
[9] Nagel, R. and Râbiger, F., Superstable operators on Banach spaces. Israel J. Math. 81(1993), 213–226. http://dx.doi.org/10.1007/BF02761307 Google Scholar
[10] Phillips, R. S., Spectral theory for semigroups of linear operators. Trans. Amer. Math. Soc. 71(1951), 393–415. http://dx.doi.org/10.1090/S0002-9947-1951-0044737-9 Google Scholar
[11] Phillips, R. S., Perturbation theory for semigroups of linear operators. Trans. Amer.Math. Soc. 74(1953), 199–221. http://dx.doi.org/10.1090/S0002-9947-1953-0054167-3 Google Scholar
[12] Râbiger, F. and Wolff, M., Superstable semigroups of operators. Indag. Math. 6(1995), 481–494. http://dx.doi.org/10.1016/0019-3577(96)81762-9 Google Scholar
[13] Voigt, J., A perturbation theorem for the essential spectral radius of strongly continuous semigroups. Monatsh. Math. 90(1980), 153–161. http://dx.doi.org/10.1007/BF01303264 Google Scholar
[14] Voigt, J., Stability of the essential type of strongly continuous semigroups. Trans. Steklov Inst. Math. 203(1994), 469–477.Google Scholar