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

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.