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Remarks on embeddable semigroups in groups and a generalization of some Cuthbert's results

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  • Khalid Latrach
  • Abdelkader Dehici

Abstract

Let ( U ( t ) ) t ≥ 0 be a C 0 -semigroup of bounded linear operators on a Banach space X . In this paper, we establish that if, for some t 0 > 0 , U ( t 0 ) is a Fredholm (resp., semi-Fredholm) operator, then ( U ( t ) ) t ≥ 0 is a Fredholm (resp., semi-Fredholm) semigroup. Moreover, we give a necessary and sufficient condition guaranteeing that ( U ( t ) ) t ≥ 0 can be imbedded in a C 0 -group on X . Also we study semigroups which are near the identity in the sense that there exists t 0 > 0 such that U ( t 0 ) − I ∈ 𝒥 ( X ) , where 𝒥 ( X ) is an arbitrary closed two-sided ideal contained in the set of Fredholm perturbations. We close this paper by discussing the case where 𝒥 ( X ) is replaced by some subsets of the set of polynomially compact perturbations.

Suggested Citation

  • Khalid Latrach & Abdelkader Dehici, 2003. "Remarks on embeddable semigroups in groups and a generalization of some Cuthbert's results," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2003, pages 1-11, January.
    • Handle: RePEc:hin:jijmms:580140
    DOI: 10.1155/S0161171203011839
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