Stud. Univ. Babe¸ s-Bolyai Math. 66(2021), No. 2, 297–305 DOI: 10.24193/subbmath.2021.2.06 Barbashin conditions for uniform instability of evolution operators Mihail Megan and Rovana Boruga (Toma) Dedicated to Professor Gheorghe Coman on the occasion of his 85th anniversary. Abstract. The aim of the present paper is to give some characterization theorems of Barbashin type for the uniform exponential instability and uniform polynomial instability behavior of evolution operators. Also, some examples which illustrate the connections between the concepts presented are given. Mathematics Subject Classification (2010): 47B01, 34D05. Keywords: Evolution operator, uniform instability, Barbashin conditions. 1. Introduction In the last period significant progress has been made in the study of exponential stability, dichotomy and trichotomy in Banach spaces. A great number of papers that describe the asymptotic behavior of evolution operators in the exponential case was published, see for example [7], [8], [9], [11] and the references therein. In particular, the uniform exponential instability was studied in [5], [4], [12], [13], [15]. Later, the need for a new approach arose from the fact that in some situations, in particular for nonautonomous systems, the exponential stability is too stringent. In this sense a polynomial asymptotic behavior was introduced by L. Barreira and C. Valls ([2]) for the continuous case, respectively by A.J.G. Bento and C.M.Silva ([3]) for discrete-time systems. Also, another interesting idea in this area can be found in [10] where A.L. Sasu, M. Megan and B. Sasu give some theorems of characterization for the concept of uniform exponential instability in terms of Banach function spaces. Recently, the same authors proposed in [14] an overview in the framework of Banach sequence spaces and their applications in the asymptotic theory of variational equations. In this paper we focus on the concepts of uniform exponential instability, ∗- uniform exponential instability, uniform polynomial instability and ∗-uniform poly- nomial instability for evolution operators in Banach spaces.