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Zbl 1069.47051
Latrach, Khalid; Dehici, Abdelkader
Remarks on embeddable semigroups in groups and a generalization of some Cuthbert's results. (English)
[J] Int. J. Math. Math. Sci. 2003, No. 22, 1421-1431 (2003). ISSN 0161-1712; ISSN 1687-0425/e

The paper consists of two parts which could be viewed as independent of each other, except for two embedding statements (i.e., Remark 3.3 and Proposition 4.2) which both refer to Theorem 2.1 of the first part (cf. my comments below). According to Theorem 2.1 and Proposition 2.2, every $C_0$-semigroup of bounded linear operators on a Banach space $X$ can be embedded in a Fredholm semigroup: a rather surprising result! However, the reason why I am emphasizing the independence is based on the observation that seemingly the authors' proof of Theorem 2.1 and the related Proposition 2.2 are incomplete. A careful look at the authors' proof of Proposition 2.2 reveals that it seems to be not clear that $U(t_1)x \not= 0$. Therefore, it seems to be not clear that $0$ is an eigenvalue of $U(t_1)$, as claimed by the authors, implying that the inductive construction of the non-empty (!) sets $\Lambda_n$ seemingly cannot be accepted in its present state. Concerning the second part of the paper under review, the authors transfer results of J. R. Cuthbert from compact operators to arbitrary closed two-sided ideals ${\cal{J}}(X)$ of bounded linear operators on a ${\Bbb{C}}$-Banach space $X$ which are contained in the ideal of the so-called Fredholm perturbations (a concept which is introduced by the authors), i.\,e., they investigate the set of all $t \geq 0$ such that $U(t) - I \in {\cal{J}}(X)$, where $(U(t))$ is a given $C_0$-semigroup on $X$, in relation to the (geometric) structure of the underlying Banach space $X$. The authors then generalize these investigations by substituting ${\cal{J}}(X)$ through the set of power compact operators.
[Frank Oertel (Edinburgh)]

MSC 2000:: *47D60 C-semigroups
46B20 Geometry and structure of normed spaces
47A10 Spectrum and resolvent of linear operators
47A53 (Semi-)Fredholm operators; index theories
47A55 Perturbation theory of linear operators
47L20 Operator ideals

Keywords: semigroup; Fredholm operators; Fredholm perturbation; operator ideals

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