Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

If You Use a Screen Reader

This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.

Lack of Uniform Stabilization for Noncontractive Semigroups Under Compact Perturbation

R. Triggiani
Proceedings of the American Mathematical Society
Vol. 105, No. 2 (Feb., 1989), pp. 375-383
DOI: 10.2307/2046953
Stable URL: http://www.jstor.org/stable/2046953
Page Count: 9
  • Read Online (Free)
  • Download ($30.00)
  • Subscribe ($19.50)
  • Cite this Item
Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Lack of Uniform Stabilization for Noncontractive Semigroups Under Compact Perturbation
Preview not available

Abstract

Let $G(t), t \geq 0$, be a strongly continuous semigroup on a Hilbert space $X$ (or, more generally, on a reflexive Banach space with the approximating property), with infinitesimal generator $A$. Let: (i) either $G(t)$ or $G^\ast(t)$ be strongly stable, yet (ii) not uniformly stable as $t \rightarrow +\infty$. Then, for any compact operator $B$ on $X$, the semigroup $S_B(t)$ generated by $A + B$ cannot be uniformly stable as $t \rightarrow +\infty$. This result is `optimal' within the class of compact perturbations $B$. It improves upon a prior result in [G.1] by removing the assumption that $G(t)$ be a contraction for positive times. Moreover, it complements a result in [R.1] where $G(t)$ was assumed to be a group, contractive for negative times. Our proof is different from both [R.1 and G.1]. Application include physically significant dynamical systems of hyperbolic type in feedback form, where the results of either [R.1 or G.1] are not applicable, as the free dynamics is not a contraction.

Page Thumbnails

  • Thumbnail: Page 
375
    375
  • Thumbnail: Page 
376
    376
  • Thumbnail: Page 
377
    377
  • Thumbnail: Page 
378
    378
  • Thumbnail: Page 
379
    379
  • Thumbnail: Page 
380
    380
  • Thumbnail: Page 
381
    381
  • Thumbnail: Page 
382
    382
  • Thumbnail: Page 
383
    383