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 lead––under fullness assumption––to results of the following type: Given a continuous one-parameter group (normalizing matrices) and a compact group (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 . Our aim is to provide elementary proofs of the following results—independently of the probabilistic background––which are interesting in their own right: (1) There exists a suitable power ak which is embeddable into a continuous one-parameter group belonging to the normaliser of K, and (2) there exists a shifted power b := ak • κ, κ K such that b is embeddable into a one-parameter group belonging to the centraliser of K.