When you hear the QF CSM do his "welcome aboard" PA on a trans Pacific service, he/she normally goes through a list of airlines that are codesharing on the flight. On average, Qantas flies six B744 flights per day between Oz and the US west coast, but I think that at least 2 aircraft loads would be non QF pax. American Airlines has shown no interest in operating the route themselves because they can buy seats cheaper on Qantas, thus they use their own capacity on other lucrative routes.
Despite what aviation writers say about Qantas operating 80% of the flights, they don't have 80% market share.