Spectral flow as winding number and integral formulas

Author:
Charlotte Wahl

Journal:
Proc. Amer. Math. Soc. **135** (2007), 4063-4073

MSC (2000):
Primary 58J30; Secondary 47B10

Published electronically:
September 12, 2007

MathSciNet review:
2341959

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A general integral formula for the spectral flow of a path of unbounded selfadjoint Fredholm operators subject to certain summability conditions is derived from the interpretation of the spectral flow as a winding number.

**[APS]**M. F. Atiyah, V. K. Patodi, and I. M. Singer,*Spectral asymmetry and Riemannian geometry. III*, Math. Proc. Cambridge Philos. Soc.**79**(1976), no. 1, 71–99. MR**0397799****[ASS]**J. Avron, R. Seiler, and B. Simon,*The index of a pair of projections*, J. Funct. Anal.**120**(1994), no. 1, 220–237. MR**1262254**, 10.1006/jfan.1994.1031**[BCPRSW]**Moulay-Tahar Benameur, Alan L. Carey, John Phillips, Adam Rennie, Fyodor A. Sukochev, and Krzysztof P. Wojciechowski,*An analytic approach to spectral flow in von Neumann algebras*, Analysis, geometry and topology of elliptic operators, World Sci. Publ., Hackensack, NJ, 2006, pp. 297–352. MR**2246773****[BLP]**Bernhelm Booss-Bavnbek, Matthias Lesch, and John Phillips,*Unbounded Fredholm operators and spectral flow*, Canad. J. Math.**57**(2005), no. 2, 225–250. MR**2124916**, 10.4153/CJM-2005-010-1**[CP1]**Alan Carey and John Phillips,*Unbounded Fredholm modules and spectral flow*, Canad. J. Math.**50**(1998), no. 4, 673–718. MR**1638603**, 10.4153/CJM-1998-038-x**[CP2]**Alan Carey and John Phillips,*Spectral flow in Fredholm modules, eta invariants and the JLO cocycle*, 𝐾-Theory**31**(2004), no. 2, 135–194. MR**2053481**, 10.1023/B:KTHE.0000022922.68170.61**[Gl]**Ezra Getzler,*The odd Chern character in cyclic homology and spectral flow*, Topology**32**(1993), no. 3, 489–507. MR**1231957**, 10.1016/0040-9383(93)90002-D**[GS]**Ezra Getzler and András Szenes,*On the Chern character of a theta-summable Fredholm module*, J. Funct. Anal.**84**(1989), no. 2, 343–357. MR**1001465**, 10.1016/0022-1236(89)90102-X**[KL]**Paul Kirk and Matthias Lesch,*The 𝜂-invariant, Maslov index, and spectral flow for Dirac-type operators on manifolds with boundary*, Forum Math.**16**(2004), no. 4, 553–629. MR**2044028**, 10.1515/form.2004.027**[L]**Matthias Lesch,*The uniqueness of the spectral flow on spaces of unbounded self-adjoint Fredholm operators*, Spectral geometry of manifolds with boundary and decomposition of manifolds, Contemp. Math., vol. 366, Amer. Math. Soc., Providence, RI, 2005, pp. 193–224. MR**2114489**, 10.1090/conm/366/06730**[Wa]**C. Wahl, ``A new topology on the space of unbounded selfadjoint operators and the spectral flow'', preprint math.FA/0607783 on arXiv (2006).

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
58J30,
47B10

Retrieve articles in all journals with MSC (2000): 58J30, 47B10

Additional Information

**Charlotte Wahl**

Affiliation:
Mathematisches Inst., Georg-August-Universität Göttingen, Bunsenstr. 3-5, 37073 Göttingen, Germany

Email:
ac.wahl@web.de

DOI:
https://doi.org/10.1090/S0002-9939-07-08919-8

Keywords:
Spectral flow,
integral formula,
winding number,
Schatten ideal

Received by editor(s):
July 5, 2006

Received by editor(s) in revised form:
September 11, 2006

Published electronically:
September 12, 2007

Communicated by:
Mikhail Shubin

Article copyright:
© Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.