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

In this paper we study nonuniform dichotomy concepts of linear evolutionary processes which are defined in a general Banach space and whose norms can increase no faster than an exponential. Connections between the dichotomy concepts and (B, D)admissibility properties are established. These connections have been partially accomplished in an earlier paper by the authors for the case when the process was a semigroup of class C and (B, D) = [I?, L«) . IntroductionThe Thus this paper is in a sense a sequel to [5]. Notation, definitions and terminologyLet ( The space of X-valued functions / almost defined on IR such that / is strongly measurable and locally integrable is denoted by L (X) . In particular L, (R) = £. . If I = [a, b] is a real compact loc locinterval, then the characteristic function of J will be denoted by cp .In the particular cases a = 0 and respectively b = °° we use the Throughout in this paper we suppose that for all D € S and t > 0 the set is a closed complemented subspace. 36P e t r e P r e d a a n d M i h a i I MeganKer P 1 (* Q ) = ^0 0 ) ) a n d w e l e t p 2 (* 0 ) = J " P 1^0^( w h e r e J i s t h e

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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