Functional Analysis and Applications Seminar 

31/10/2008, 15:00 — 16:00 — Room P3.10, Mathematics Building
Catarina Carvalho, Instituto Superior Técnico, U.T. Lisboa

Topological index for pseudodifferential operators on $\mathbb{R}^n$

We develop a pseudodifferential calculus on $\mathbb{R}^n$ suitable to the study of the Fredholm index from the topological viewpoint, namely to generalizations of the index formula on compact spaces. We consider operators that are multiplication by a matrix valued function outside a compact set and give some properties of this class. We then study the closure of a suitable subalgebra. In particular, we find explicit Fredholm criteria and show that the symbols of Fredholm operators have similar topogical properties to those on compact spaces, leading in the same way to a topological formula for the index. Our results have applications on wider classes of pseudodifferential operators on $\mathbb{R}^n$ , namely the isotropic and scattering calculi, and also to pseudodifferential operators on non-compact manifolds.