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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: booss@mmf.ruc.dk
Matthias Lesch
Affiliation:
Mathematisches Institut, Universität zu Köln, Weyertal 86–90, D–50931 Köln, Germany, e-mail: lesch@mi.uni-koeln.de
John Phillips
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, V8W 3P4, e-mail: phillips@math.uvic.ca
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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

References

[1] Atiyah, M. F. and Singer, I. M., Index theory for skew-adjoint Fredholm operators. Inst. Hautes Études Sci. Publ. Math. 37(1969), 5–26 Google Scholar
[2] Booss-Bavnbek, B., Unique continuation property for Dirac operators, revisited. Contemp.Math. 258(2000), 21–32.Google Scholar
[3] Booss-Bavnbek, B. and Furutani, K., The Maslov index: a functional analytical definition and the spectral flow formula. Tokyo J. Math. 21(1998), 1–34.Google Scholar
[4] Booss-Bavnbek, B., Lesch, M. and Phillips, J., Spectral flow of paths of self–adjoint Fredholm operators. Nuclear Physics B 104(2002), 177–180.Google Scholar
[5] Booss-Bavnbek, B. and Wojciechowski, K. P., Elliptic Boundary Problems for Dirac Operators. Birkhäuser, Boston, (1993).Google Scholar
[6] Cordes, H. O. and Labrousse, J. P., The invariance of the index in the metric space of closed operators. J. Math. Mech. 12(1963), 693–719.Google Scholar
[7] Douglas, R. G. and Wojciechowski, K. P., Adiabatic limits of the η-invariants. The odd-dimensional Atiyah–Patodi–Singer problem. Comm.. Math. Phys. 142(1991), 139–168.Google Scholar
[8] Dunford, N. and Schwarz, J. T., Linear Operators. Interscience Publication, 1958. ReprintedWiley Classics Library, 1988.Google Scholar
[9] Furutani, K. and Otsuki, N., Maslov index in the infinite dimension and a splitting formula for a spectral flow. Japan J. Math. 28(2003), 215–243.Google Scholar
[10] Grubb, G., Trace expansions for pseudodifferential boundary problems for Dirac-type operators and more general systems. Ark.Mat. 37(1999), 45–86.Google Scholar
[11] Himpel, B., Kirk, P. and Lesch, M., Calderón projector for the Hessian of the perturbed Chern-Simons function on a 3-manifold with boundary. Proc. LondonMath. Soc. 3 89(2004), 241–272.Google Scholar
[12] Kato, T., Perturbation Theory for Linear Operators. Second edition. Grundlehren der MathematischenWissenschaften 132, Springer–Verlag, Berlin, 1976.Google Scholar
[13] Kirk, P. and Lesch, M., The η-invariant, Maslov index, and spectral flow for Dirac-type operators on manifolds with boundary. ForumMath. 16 (2004), 553–629.Google Scholar
[14] Lesch, M., The uniqueness of the spectral flow on spaces of unbounded self-adjoint Fredholm operators. Contemp.Math. 366(2005), 193–224.Google Scholar
[15] Nicolaescu, L., On the space of Fredholm operators. math.DG/0005089 (2000).Google Scholar
[16] Phillips, J., Self-adjoint Fredholm operators and spectral flow. Canad. Math. Bull. 39(1996), 460–467.Google Scholar
[17] Riesz, F. and Nagy, B., Functional Analysis. Dover Publications, New York, 1990.Google Scholar
[18] Robbin, J. and Salamon, D., The spectral flow and the Maslov index. Bull. LondonMath. Soc. 27(1995), 1–33.Google Scholar
[19] Rudin, W., Functional Analysis. Second edition. McGraw-Hill, New York, 1991.Google Scholar