COMPACT-OPERATORS ON A BANACH-SPACE INTO THE SPACE OF ALMOST PERIODIC-FUNCTIONS

Let S be a topological semigroup and AP(S) the space of continous complex almost periodic functions on S. We obtain characterizations of compact and weakly compact operators from a Banach space X into AP(S). For this we use the almost periodic compactification of S obtained through uniform spaces. For a bounded linear operator T from X into AP(S), let T 5, be the translate of T by s in S defined by T 5(x)=(Tx) 5 . We define topologies on the space of bounded linear operators from X into AP(S) and obtain the necessary and sufficient conditions for an operator T to be compact or weakly compact in terms of the uniform continuity of the map s→T 5. If S is a Hausdorff topological semigroup, we also obtain characterizations of compact and weakly compact multipliers on AP(S) in terms of the uniform continuity of the map S→μs, where μs denotes the unique vector measure corresponding to the operator T 5. © 1990 Indian Academy of Sciences.