Index theory for unbounded Fredholm operators IIOberseminar Global Analysis and Operator Algebras

by Marina Prokhorova (MPIM, University of Haifa & Technion)

Tuesday Dec 9, 2025, 2:15 PM → 3:45 PM Europe/Berlin

Endenicher Allee 60/1-008 (Mathezentrum)

In continuation of my previous talk two weeks ago, I will show other approaches to
index theory for families of unbounded Fredholm operators in a Hilbert space. I will
start by recalling basic notions for those who were not present at the first talk.
Then I will show how one can modify a graph continuous family by an appropriate
“change of coordinates”, giving rise to a norm continuous family of Fredholm
operators. For self-adjoint operators, the situation is more complicated; I will
describe arising obstructions and show that such an “improving of topology” can
still be achieved in most important cases. In particular, these results explain why
a proof of existence of spectral sections by Melrose and Piazza is working for
fibrations. The talk is based on my preprint arXiv:2202.03337. I will also briefly
present a recent approach of N.V. Ivanov to index theory of families
(arXiv:2111.15081), which works even under much weaker continuity assumptions.