Spectre operator, achievement sets and sets of P-sums in a hyperspace of compact sets

Let $(X,+,d)$ be an Abelian metric group and A⊂ X. We investigate the spectre of a set A, defined as the set of all elements z∈ X such that for every x∈ A either x+z ∈ A or x-z ∈ A. We consider the corresponding to this notion operator S acting on the hyperspace of compact sets and examine its properties. Furthermore, we study the families of achievement sets and sets of P-sums in this hyperspace, as well as prove some properties of achievement sets in the plane.