Index theory for unbounded Fredholm operatorsOberseminar Global Analysis and Operator Algebras
by
Endenicher Allee 60/1-008
Mathezentrum
As was shown in classical works of Atiyah, Jänich, and Singer, the space of bounded Fredholm operators represents even K-theory, while its subspace consisting of self-adjoint operators (more precisely, its non-trivial connected component) represents odd K-theory. The index theory of elliptic differential operators on closed manifolds is based on these classical results.
However, in some situations, e.g. for elliptic operators on manifolds with boundary, one needs to deal with families of unbounded operators. My talk is devoted to an index theory of such families. I will explain how relevant spaces of unbounded operators are related to classical spaces of bounded Fredholm operators and show that natural maps between them are homotopy equivalences. In addition, I will answer a question raised in 2001 by Booss-Bavnbek, Lesch, and Phillips.
The talk is based on my preprint arXiv:2110.14359 . If time permits, I will also mention a related result of arXiv:2202.03337 .