Normalisers and centralisers of compact matrix groups. An elementary approach

Abstract

Early investigations of operator stable laws and operator self-similar stochastic processes on a finite-dimensional vector space V = R d 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. ( E c := 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 { a k : 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 a k 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 := a k · κ, κ ∈ K such that b is embeddable into a one-parameter group { exp ( t · E c ) : t ∈ R } belonging to the centraliser of K.