Unbounded Fredholm Operators and Spectral Flow

2001/08/31 by Bernhelm Booss-Bavnbek, Matthias Lesch, John Phillips · 3 citations
Mathematics · #math.FA #math.OA #math.SP #msc:58J30 #msc:47A53 #msc:19K56 #msc:58J32

paper · pdf

published as Canadian Journal of Mathematics vol. 57, no.2 (2005), 225-250. · 23 pages, 2 figures; 09/10/2001 minor corrections, Proposition characterizing the range of the Riesz transformation added; 02/12/2004 very final version 1.0.2, minor corrections

arxiv created 2004/02/12 · arxiv updated 2009/11/30

Abstract

We study the gap (= "projection norm" = "graph distance") topology of the space of (not necessarily bounded) self--adjoint Fredholm operators in a separable Hilbert space by the Cayley transform and direct methods. In particular, we show that the space is connected contrary 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.

Cited by

• Index theory for non-compact quantum graphs
• On the Uniqueness of the G-equivariant Spectral Flow
• Abstract cone operators and Lipschitz rigidity for scalar curvature on singular manifolds

Related

• A new topology on the space of unbounded selfadjoint operators and the spectral flow
• Ordinary differential equations in Banach spaces and the spectral flow
• On a spectral flow formula for the homological index
• Spectral flow and winding number in von Neumann algebras
• On the spectral flow for paths of essentially hyperbolic bounded operators on Banach spaces