Characterisations of compact operators on the space of almost periodic functions

Abstract

Let X be a Banach space and AP(IR, X) the space of continuous almost periodic functions on IR: into X with supremum norm. We obtam characterisations of compact operators on AP(IR, X) (Theorem 3.1) This theorem is then used to prove the following Let AP(IR) be the space AP(IR, C). For a compact operator K on AP(IM.), the aggregate of Fourier exponents of functions of the range of K is countable even though the range of K is uncountable. We also obtain suffiCIent conditions on K so that the Fourier series of all the functions in the range of K converges at the same point.