Normalisers and centralisers of compact matrix groups. An elementary approach

AbstractEarly 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:(1)There exists a suitable power ak which is embeddable into a continuous one-parameter group {exp(t·E):t∈R} 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 {exp(t·Ec):t∈R} belonging to the centraliser of K