Abstract We study some problems concerning derivatives and indefinite integrals of elements of wide classes of functions (including almost periodic, recurrent, almost automorphic and Eberlein almost periodic functions). We introduce a notion of a spectrum of a function with respect to a class of functions. This notion enables us to investigate almost periodicity of the bounded solutions of certain functional equations.
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Let J be an abelian topological semigroup and C a subset of a Banach space X. Let L(X) be the space of bounded linear operators on X and Lip(C) the space of Lipschitz functions ⨍: C → C. We exhibit a large class of semigroups J for which every weakly continuous semigroup homomorphism T: J → L(X) is necessarily strongly continuous. Similar results are obtained for weakly continuous homomorphisms T: J → Lip(C) and for strongly measurable homomorphisms T: J → L(X).