CARPATHIAN J. MATH. 37 (2021), No. 1, 45 - 51 Online version at https://carpathian.cunbm.utcluj.ro Print Edition: ISSN 1584 - 2851 Online Edition: ISSN 1843 - 4401 Datko type characterizations for nonuniform polynomial dichotomy ROVANA BORUGA (TOMA) and MIHAIL MEGAN ABSTRACT. The aim of the present paper is to give two characterization theorems of Datko type for the nonuniform polynomial dichotomy concept with respect to invariant projection families and also with respect to strongly invariant projection families. 1. I NTRODUCTION In the last decades, one of the most important topics discussed in the field of dynamical systems is the uniform exponential dichotomy behavior of evolution operators in Banach spaces. This concept was mentioned for the first time in the work of O. Perron [14] and refers to the fact that the state space can be decomposed in every moment into a direct sum of two subspaces: the stable subspace and the unstable subspace. Over the years, the uniform exponential dichotomy notion has been generalized in various forms and researchers dedicated their study to the nonuniform case [8], [9], [12], [13], [16]. The starting point in studying this type of behavior is in the work [2] of the authors L. Barreira and C. Valls who had a notable contribution in developing it. Also, in this direction it is important to mention an interesting paper of D. Dragiˇ cevi´ c [5] where the author deals with the nonuniform exponential behavior. More precisely, he obtains some Datko-type characterizations of several classes of a strong nonuniform exponential behavior: contractions, expansions and dichotomies, for both continuous and discrete time cases. Recently, N. Lupa and L.H.Popescu [7], being motivated by the work of Dragiˇ cevi´ c [5], generalize Datko’s theorem on a class of admissible Banach function spaces, using completely different type of techniques from those in [5]. Moreover, in order to generalize the exponential bahavior, L. Barreira and C. Valls [1] introduced in 2009 a more general behavior, namely the nonuniform polynomial di- chotomy, which they studied for the continuous case. They were followed by A. J. G. Bento and C. M. Silva [3] who considered the same concept for discrete time systems. Other results in this area were obtained in the papers [4], [10], [11], [15]. In this paper we obtain necessary and sufficient conditions of Datko type for the nonuni- form polynomial dichotomy concept of evolution operators regarding two types of pro- jections families: invariant and respectively strongly invariant projection families to the evolution operator. 2. NOTATIONS AND DEFINITIONS Let X be a real or complex Banach space and B(X) the Banach algebra of all bounded linear operators acting on X. We will denote by ‖.‖ the norm on X and on B(X) and let I be the identity operator on X. We also consider Δ and T two sets defined by Received: 16.06.2020. In revised form: 16.12.2020. Accepted: 23.12.2020 2010 Mathematics Subject Classification. 34D05, 34D09. Key words and phrases. Nonuniform polynomial dichotomy, invariant projections, strongly invariant projections. Corresponding author: Rovana Boruga(Toma); rovanaboruga@gmail.com 45