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Zbl 1058.34072
Huy, Nguyen Thieu
Exponentially dichotomous operators and exponential dichotomy of evolution equations on the half-line. (English)
[J] Integral Equations Oper. Theory 48, No. 4, 497-510 (2004). ISSN 0378-620X; ISSN 1420-8989/e

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 {\it N.~V.~Minh}, {\it F. Räbiger} and {\it 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 $U(t,s)$ was characterized in terms of inversion properties of a suitably defined operator, $I_Z$, associated to the solutions of the integral equation $$ u(t) = U(t,s)\,u(s) + \int_s^t U(t,\xi)\,f(\xi)\,d\xi, \qquad t \geq s \geq 0. $$ 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. \par The new result allows the author to extend to a general Banach space results proved for finite-dimensional spaces by {\it A. Ben-Artzi}, {\it I. Gohberg} and {\it M. A. Kaashoek} [J.\ Dyn.\ Differ.\ Equations 5, 1-36 (1993; Zbl 0771.34011)].
[Luigi Barletti (Firenze)]

MSC 2000:: *34G10 Linear ODE in abstract spaces
34D09 Dichotomy, trichotomy
47D06 One-parameter semigroups and linear evolution equations

Keywords: exponential dichotomy; evolution families; exponentially dichotomous operators

Citations: Zbl 0977.34056; Zbl 0771.34011

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