Anders Karlsson: Metric functional analysis, or why every isometry has a fixed point

Location

FR4, Albanova

Schedule

14:15–15:00 Pre-colloquium by Tim Schmatzler  in FA32.

15:15–16:15 Colloquium lecture by Anders Karlsson.
16:15–17:00 SMC social get together with refreshments.

Abstract

Fixed point theorems, such as those of Brouwer and Banach, play a fundamental role throughout the mathematical sciences. In this talk I will explain a new fixed point theorem asserting that every isometry of a metric space has a fixed point, though not necessarily inside the space itself, but in a natural compactification of it.

This result is new even for Banach spaces, and in that setting it yields a new proof of the classical von Neumann–Carleman mean ergodic theorem. The classical formulation of the mean ergodic theorem is well-known to fail in general Banach spaces, but the new one always holds true. Another consequence is that every invertible bounded linear operator of a Hilbert space admits a nontrivial invariant metric functional — the metric space analogue of a linear functional — on the space of positive operators.

The notion of metric functionals leads to a metric theory in analogy with functional analysis. This leads to extensions of for example the Denjoy–Wolff theorem in complex dynamics and Thurston's spectral theorem for surface homeomorphisms.