Normalisers and centralisers of compact matrix groups. An elementary approach

Early investigations of operator stable laws and operator self-similar stochastic processes on a finite-dimensional vector space V=Rd lead--under fullness assumption--to results of the following type: Given a continuous one-parameter group {exp(t·E):t∈R}⊆GL(V) (normalizing matrices) and a compact group K⊆GL(V) (symmetries) such that exp(t · E) normalizes K for all t, then there exists a modification (exp(t · (E + H)) = {exp(t · E) · exp(t · H)} centralizing K where {exp(t · H)} ⊆ K. (Ec := E + H is called commuting exponent.) It is challenging to obtain similar results in the context of operator semistable laws or operator semi-self-similar processes where the continuous one-parameter matrix group is replaced by a discrete group {ak:k∈Z}. Our aim is to provide elementary proofs of the following results-independently of the probabilistic background--which are interesting in their own right: