ADMISSIBILITY VERSUS NONUNIFORM EXPONENTIAL BEHAVIOR FOR NONINVERTIBLE COCYCLES

We study the relation between the notions of exponential dichotomy and admissibility for a nonautonomous dynamics with discrete time. More precisely, we consider Z-cocycles defined by a sequence of linear operators in a Banach space, and we give criteria for the existence of an exponential dichotomy in terms of the admissibility of the pairs (l(p), l(q)) of spaces of sequences, with p <= q and (p, q) not equal (1, infinity). We extend the existing results in several directions. Namely, we consider the general case of nonuniform exponential dichotomies; we consider Z-cocycles and not only N-cocycles; and we consider exponential dichotomies that need not be invertible in the stable direction. We also exhibit a collection of admissible pairs of spaces of sequences for any nonuniform exponential dichotomy.