General almost automorphy

Abstract

We will propose a very general setting for the following proposition. I a linear operator Lf is almost automorphic and all bounded solutions of the homogeneous equation Lg = 0 are almost automorphic then, for almost automorphic so, all bounded solutions of the inhomogeneous equation Lf = so are also almost automorphic.The almost automorphy previously introduced by us will be generalized and diversified into a property which we will name "A-invariance"; and we will deal with both strong solutions and weak solutions. We will also compare our A-invariance with minimality, and propose a rather thinned-out almost periodicity that will be associated with our A-invariance.We deal throughout with convergence by sequences and not by general nets, and for the sake of weak solutions, which will require a Lagrange adjoint operator L*X(t), we will favor a Euclidean setting. But strong solutions can be easily presented in an abstract setting, and we will begin with such a one. I. Strong solutionsWe stipulate the following data.(a) A linear vector space F over real or complex numbers with a "sufficient" number of pseudo-norms.(b) A partially ordered index set {a;<j, and a family tAs1 of "translations" of parts of F into F of the following kind. Given an element f E F, for any index 5' there is a "lesser" index 6, 5 < ', such that A j is defined. Thus, for every semi-norm, IIgI < 1/n2 11f11 for all n, so that jjg = 0. Thusg = 0,orAbf=f.Remark 1: Axiomatically, [4] leads to g = 0 by the following assumption, without the use of semi-norms or any other topology on F. With each f e F there is associable a subset U of F which contains f and all its translates Abnf, such that the set V = U-U is "bounded in every direction", in the sense that no element g # 0 can be contained in V with all its multiples ng. II. Euclidean settingWe realize the data of I. in the following way.(a) On a Euclidean RN, F consists of all functions which are CO-bounded, that is, which are defined on RN and are bounded and uniformly continuous throughout. And we endow them with the sup-norm l/ = suplf(t)I, t C RN[5] (b) Given f(t) E RN, any infinite sequence a' = {an'} in RN, in which repetitions are allowed, contains an infinite subsequence a = lan} , in symbols a < a', for which the translation operator Tjf = lim n-f(t + an) = g(t) [6] exists. The limit function g(t) is again CO-bounded; and since a subsequence of a subsequence is a subsequence, the operator [6] can be iterated, so that TaTaf, T3aTTaT7af, etc., can be formed. Also, two "independent" sequences a',f3' contain subsequences af, for which product operators like TflTaf TaTi-flf,