Alaa E. Hamza and Gilbert L. Muraz

Abstract

         Let X be a complex Banach space and let M be a closed subspace of  L^∞(J,X), where J Î {R, R⁺}. We answer the following question: Under what conditions Φ_s - Φ Î M, " s Î J implies that Φ Î M. Some conditions will be imposed on M to obtain the main result concerning the indefinite integral. These conditions guarantee the following implication :  F Î E(J,X)  -----> F Î M , where F is the integral  \int_0^t f(s)ds of  f Î M ∩ C_ub(J,X) . Also, we generalize Loomis' Theorem for almost periodic functions [19, Theorem 5], to a more general class of functions M Í L^∞(R,X) containing AP(R,X). The main result of Part IV is: If Φ is uniformly continuous, bounded, such that the M-spectrum σ_M(Φ) of Φ is at most countable and, for every λÎ σ_M(Φ), the function e^-iλtΦ(t)  is ergodic, then  Φ Î M.