*Title : The L1-Karcher mean of positive operators
*Speaker : Palfia.Miklos
*Place / Date : 32356A / 20 October 2017 (Friday) am 10:00-12:00
*Abstract :
We study the Karcher equation corresponding to probability measures on the Borel sigma algebra of positive operators on a Hilbert space with the Thompson metric. Using the fundamental Wasserstein contraction property of the barycentric operator, also known as Karcher mean, we develop an ODE theory for the Karcher equation of L1 probability measures, in order to establish existence and uniqeness of its solution. We establish the existence of the stationary point by approximating in the Wasserstein distance an L1 probability measure by finitely supported measures. We investigate a Trotter-Kato type product formula in this setting, leading to a continuous-time law of large numbers for the ODE flow. En route we prove the norm convergence conjecture of power means to the archer mean.