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Exponentially dichotomous operators and exponential dichotomy of evolution equations on the half-line. (English) Zbl 1058.34072

The author gives a new characterization of evolution families of operators in Banach spaces that have exponential dichotomy. The framework is similar to that of N. V. Minh, F. Räbiger and R. Schnaubelt [Integral Equations Oper. Theory 32, 332-353 (1998; Zbl 0977.34056)], based on the so-called “evolution semigroup” associated to an evolution family. In that paper, the exponential dichotomy of an evolution family $U\left(t,s\right)$ was characterized in terms of inversion properties of a suitably defined operator, ${I}_{Z}$, associated to the solutions of the integral equation

$u\left(t\right)=U\left(t,s\right)\phantom{\rule{0.166667em}{0ex}}u\left(s\right)+{\int }_{s}^{t}U\left(t,\xi \right)\phantom{\rule{0.166667em}{0ex}}f\left(\xi \right)\phantom{\rule{0.166667em}{0ex}}d\xi ,\phantom{\rule{2.em}{0ex}}t\ge s\ge 0·$

In the paper under consideration, the characterization is still given in terms of the operator ${I}_{Z}$ but a different characterizing property is considered which is based on the concepts of “exponentially dichotomous” and “quasi-exponentially dichotomous” operators, the latter being introduced by the author.

The new result allows the author to extend to a general Banach space results proved for finite-dimensional spaces by A. Ben-Artzi, I. Gohberg and M. A. Kaashoek [J. Dyn. Differ. Equations 5, 1-36 (1993; Zbl 0771.34011)].

##### MSC:
 34G10 Linear ODE in abstract spaces 34D09 Dichotomy, trichotomy 47D06 One-parameter semigroups and linear evolution equations