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The aim of the present paper is to emphasize some growth concepts for the dichotomic behavior of evolution operators in Banach spaces. In fact, we approach the exponential growth, the polynomial growth and the h-growthfor both uniform and nonuniform cases. Connections between concepts are established. Majorization criteria for the uniform behaviors are given.

The paper considers a general concept of nonuniform (h, k)-dichotomy for evolution operators in Banach spaces. Two characterizations of this concept in terms of some families of norms compatible with the dichotomy projectors are given.

The paper considers the general concept of uniform exponential splitting as a generalization of uniform exponential dichotomy property for evolution operators in Banach spaces.

The paper considers some concepts of (h, k)-dichotomy and (h, k)-trichotomy for noninvertible evolution operators in Ba- nach 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, charac- terizations for nonuniform exponential trichotomy and nonuniform polynomial trichotomy are obtained.