Proximity Between Selfadjoint Operators and Between Their Associated Random Measures

Abstract

We study how the proximity between twoselfadjoint bounded operators, measured by a classical distance, canbe expressed by a proximity between the associated spectralmeasures. This last proximity is based on a partial order relation on the set of projectors. Assuming an hypothesis of commutativity, we show that the proximity between operators implies the one between theassociated spectral measures, and conversally, the proximity between spectral measures implies the one between associated selfadjoint operators.