Unbounded Fredholm Operators and Spectral Flow

[摘要] We study the gap (= ``projection norm'' = ``graph distance'') topologyof the space of all (not necessarily bounded) self-adjoint Fredholmoperators in a separable Hilbert space by the Cayley transform anddirect methods. In particular, we show the surprising result thatthis space is connected in contrast to the bounded case. Moreover, wepresent a rigorous definition of spectral flow of a path of suchoperators (actually alternative but mutually equivalent definitions)and prove the homotopy invariance. As an example, we discuss operatorcurves on manifolds with boundary.

[学科分类] 数学（综合）