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# Unbounded Fredholm Operators and Spectral Flow

Published:2005-04-01
Printed: Apr 2005
• Bernhelm Booss-Bavnbek
• Matthias Lesch
• John Phillips
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## Abstract

We study the gap (= projection norm'' = graph distance'') topology of the space of all (not necessarily bounded) self-adjoint Fredholm operators in a separable Hilbert space by the Cayley transform and direct methods. In particular, we show the surprising result that this space is connected in contrast 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.
 MSC Classifications: 58J30 - Spectral flows 47A53 - (Semi-) Fredholm operators; index theories [See also 58B15, 58J20] 19K56 - Index theory [See also 58J20, 58J22] 58J32 - Boundary value problems on manifolds