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 present paper treats a concept of (h,k)-dichotomy for linear discrete systems. Sufficient conditions for the k-boundedness of the projection sequences that give the dichotomy are presented and an illustrative example shows the connection between the growth of the system and the bound of the sequence of projections. Thus the growth of the system that is assumed in the theorems is essential.

In this paper we consider three concepts of uniform exponential trichotomy on the half-line in the general framework of evolution operators in Banach spaces. We obtain a systematic classification of uniform exponential trichotomy concepts and the connections between them.