Anders Karlsson

An isometry fixed-point theorem and applications to bounded linear operators

I will present a fixed-point theorem stating that every isometry of a metric space admits a fixed-point in the metric compactification of the injective hull of the space. When the metric space has a conical bicombing—as is the case for convex subsets of Banach spaces, CAT(0) spaces, injective spaces, or spaces of positive operators—the injective hull is not needed.

As a consequence, it yields a generalization of the von Neumann mean ergodic theorem from Hilbert spaces to arbitrary Banach spaces, in contrast to the usual mean ergodic formulation which is known to fail in general. Another corollary asserts that every invertible bounded linear operator admits an invariant metric functional on the space of positive operators. There are also consequences for biholomorphisms and diffeomorphisms.