A new characterization of exponential dichotomy for evolutionary processes in terms of (L p , L q )-admissibility is presented, using a direct treatment, without the so-called evolution semigroup. (2000). 34D05, 47D06, 93D20. Mathematics Subject Classification

A very general characterization of exponential dichotomy for evolutionary processes in terms of the admissibility of some pair of spaces which are translation invariant (the so-called Schäffer spaces) is given. It includes, as particular cases, many interesting situations among which we note the results obtained by N. van Minh, F. Räbiger and R. Schnaubelt and the authors concerning the connections between admissibility and dichotomy.

We study the exponential stability of linear skew-product semiflows on locally compact metric space with Banach fibers. Our main tool is the admissibility of a pair of the so-called Schäffer spaces. This characterization is a very general one, it includes as particular cases many interesting situations among them we can mention some results due to Clark, Datko, Latushkin, van Minh, Montgomery-Smith, Randolph, Räbiger, Schnaubelt.

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Let π = (Φ, σ) be an exponentially bounded, strongly continuous cocycle over a continuous semiflow σ. We prove that π = (Φ, σ) is uniformly exponentially stable if and only if there exist T > 0 and c ∈ (0, 1), such that for each θ ∈ Θ and x ∈ X there exists τ θ,x ∈ (0, T ] with the property thatAs a consequence of the above result we obtain generalizations, in both continuous-time and discrete-time, of the the well-known theorems of Datko-Pazy, Rolewicz and Zabczyk for an exponentially bounded, strongly continuous cocycle over a semiflow σ. A version of the above theorems for the case of the exponential instability is also obtained.