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The aim of this paper is to study the concept of uniform exponential trisplitting for skew-product semiflow in Banach spaces. This concept is a generalisation of the well-known concept of uniform exponential trichotomy. We obtain necessary and sufficient conditions for this concept of Datko’s type. a character-isation in terms of Lyapunov functions is provided. The results are obtained from the point of view of the projector families, i.e. invariant and strongly invariant.

Abstract. The paper considers some concepts of (h, k)-dichotomy and (h, k)-trichotomy for noninvertible evolution operators in Banach spaces. A characterization of the (h, k)-trichotomy of an evolution operator in terms of (h, k)-dichotomy for two associated evolution operators is given. As applications of this result, characterizations for nonuniform exponential trichotomy and nonuniform polynomial trichotomy are obtained.

"The paper treats two concepts of uniform polynomial trichotomy for the skew-evolution semi- flows in Banach spaces. We obtain the connection between them, a characterization for a property of uniform polynomial growth and a sufficient criteria for the uniform polynomial trichotomy."

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