QLunch: Frank Hansen

Speaker: Frank Hansen

Title: Geodesically convex functions and multivariate operator means

Abstract: We first determine the class of Geodesically convex trace functions. This result is then used to determine large classes of multivariate operator means given as uniquely defined minimisers of Geodesically convex functions, or equivalently, as unique solutions to a critical point equation.

In the literature only concave multivariate operator means have been studied. They are all bounded from above by the arithmetic mean. The new method also gives rise to multivariate so-called hyper-means that are bounded from below by the arithmetic mean. An interesting example is the so-called entropic hyper-mean. We discover bivariate means that cannot be extended to multivariate means.