We construct a counterexample to the injectivity conjecture of Masarotto et al (2018). Namely, we construct a class of examples of injective covariance operators on an infinite-dimensional separable Hilbert space for which the Bures--Wasserstein barycentre is highly non injective -- it has a kernel of infinite dimension.

翻译：我们构造了Masarotto等（2018）单射性猜想的一个反例。具体而言，在无限维可分Hilbert空间上，我们构造了一类单射协方差算子的实例，其Bures-Wasserstein重心高度非单射——该重心的核具有无限维。