Ergodic theorems for the \(L^1\)-Karcher mean

Recently the Karcher mean has been extended to the case of probability measures of positive operators on infinite-dimensional Hilbert spaces as the unique solution of a nonlinear operator equation on the convex Banach-Finsler manifold of positive operators. Let \((\Omega ,\mu )\) $(\mathrm{\Omega },\mu )$ be a probability space, and let \(\tau :\Omega \rightarrow \Omega \) $\tau :\mathrm{\Omega }\to \mathrm{\Omega }$ be a totally ergodic map. The main result of this paper is a new ergodic theorem for functions \( F\in L^1(\Omega ,\mathbb {P})\) $F\in L^1(\mathrm{\Omega },\mathbb{P})$ , where \(\mathbb {P}\) $\mathbb{P}$ is the open cone of the strictly positive operators acting on a (separable) Hilbert space. In our result, we use inductive means to average the elements of the orbit, and we prove that almost surely these averages converge to the Karcher mean of the push-forward measure \(F_*(\mu )\) $F_{\ast }^{}(\mu )$ . From our result, we recover the strong law of large numbers and the “no dice” results proved by the third and fourth authors in the article Strong law of large numbers for the \(L^1\) $L^1$ -Karcher mean, Journal of Func. Anal. 279 (2020). From our main result, we also deduce an ergodic theorem for Markov chains with state space included in \(\mathbb {P}\) $\mathbb{P}$ .