Index theory for unbounded Fredholm operators

A linear operator on a Hilbert space is called Fredholm if it is “almost invertible” (that is, its kernel and cokernel are finite-dimensional). 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. A proper notion of continuity for such families of operators is continuity of their graphs. 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. The talk is based on my preprint
arXiv:2110.14359 (to appear in Journal of Topology and Analysis).