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Unbounded Fredholm Operators and Spectral Flow

Published online by Cambridge University Press:  20 November 2018

Bernhelm Booss-Bavnbek ,
Matthias Lesch  and
John Phillips
Bernhelm Booss-Bavnbek
Affiliation:
Institut for Matematik og Fysik, Roskilde Universitetscenter, DK–4000 Roskilde, Denmark, e-mail: [email protected]
Matthias Lesch
Affiliation:
Mathematisches Institut, Universität zu Köln, Weyertal 86–90, D–50931 Köln, Germany, e-mail: [email protected]
John Phillips
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, V8W 3P4, e-mail: [email protected]
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Abstract

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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.

Keywords

58J3047A5319K5658J32
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 57 , Issue 2 , 01 April 2005 , pp. 225 - 250
Copyright
Copyright © Canadian Mathematical Society 2005

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