The problem of determining the weakly compact or compact homomorphisms between various Banach algebras has been much studied, the study of weakly compact or compact derivations, less so. We characterize those derivations from the convolution algebra ??pZ??q to its dual that are weakly compact, providing explicit that are not compact. The first part of the thesis characterizes meaning and preliminaries which help to understanding the body of thesis. In the ????, TF-sets were introduced by Ruppert in the investigation of weakly almost periodic subsets of semigroups. If S is a semi-topological semigroup, second part of thesis investigates the structure of AP(S) and WAP(S) (the space of almost periodic or weakly almost periodic functions on S respectively).
