The uniqueness of the spectral flow on spaces of unbounded self--adjoint Fredholm operators

By M. Lesch. 2004

83 citations recorded by the source corpus.

We discuss several natural metrics on spaces of unbounded self--adjoint operators and their relations, among them the Riesz and the graph metric. We show that the topologies of the spaces of Fredholm operators resp. invertible operators depend heavily on the metric. Nevertheless we prove that in all cases the spectral flow is up to a normalization the only integer invariant of non-closed paths which is path additive and stable under homotopies with endpoints varying in the space of invertible self-adjoint operators. Furthermore we show that for certain Riesz continuous paths of self--adjoint Fredholm operators the spectral flow can be expressed in terms of the index of the pair of positive spectral projections at the endpoints. Finally we review the Cordes--Labrousse theorem on the stability of the Fredholm index with respect to the graph metric in a modern language and we generalize it to the Clifford index and to the equivariant index.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/75c91208f912e69bdc77a39ded9591fc7aaf3acc

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Bernhelm Booss-Bavnbek
2. J. Phillips
3. M. Lesch
4. Alan L. Carey
5. Fedor Sukochev

In Mathematical Physics

1. Bernhelm Booss-Bavnbek
2. J. Phillips
3. M. Lesch
4. A. Rennie
5. Alan L. Carey
6. Fedor Sukochev
7. B. Mesland

Citations in the corpus

• Unbounded Fredholm Operators and Spectral Flow (2001, background): “In [14] it is shown that SF is the only integer-valued homotopy invariant of paths in CF sa satisfying standard properties.”