Trace formulas for a class of non-Fredholm operators: A review

Authors

Alan L. Carey, University of WollongongFollow
Fritz Gesztesy, University of Missouri
Harald Grosse, University of Vienna
Galina Levitina, University of New South Wales
Denis Potapov, University of New South Wales
F Sukochev, University of New South Wales
Dima Zanin, University of New South Wales

RIS ID

114837

Publication Details

Carey, A., Gesztesy, F., Grosse, H., Levitina, G., Potapov, D., Sukochev, F. & Zanin, D. (2016). Trace formulas for a class of non-Fredholm operators: A review. Reviews in Mathematical Physics: a journal for survey and expository articles in the field of mathematical physics, 28 (10), 1630002-1-1630002-55.

Abstract

Take a one-parameter family of self-adjoint Fredholm operators {A(t)}t∈R on a Hilbert space H, joining endpoints A±. There is a long history of work on the question of whether the spectral flow along this path is given by the index of the operator DA=(d/dt)+A acting in L2(R;H), where A denotes the multiplication operator (Af)(t)=A(t)f(t) for f∈dom(A). Most results are about the case where the operators A(⋅) have compact resolvent. In this article, we review what is known when these operators have some essential spectrum and describe some new results.