Unbounded Fredholm Operators and Spectral Flow

Details Unbounded Fredholm Operators and Spectral Flow Unbounded Fredholm Operators and Spectral Flow

2001-08-02

arxiv.org/abs/math/0108014v3

Canadian Journal of Mathematics vol. 57, no.2 (2005), 225-250.

Mathematics

Functional Analysis

23 pages, 2 figures; 09/10/2001 minor corrections, Proposition characterizing the range of the Riesz transformation added; 02/

Scientific paper

We study the gap (= "projection norm" = "graph distance") topology of the space of (not necessarily bounded) self--adjoint Fredholm operators in a separable Hilbert space by the Cayley transform and direct methods. In particular, we show that the space is connected contrary to the bounded case. Moreover, we present a rigorous definition of spectral flow of a path of such operators (actually alternative but mutually equivalent definitions) and prove the homotopy invariance. As an example, we discuss operator curves on manifolds with boundary.

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