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

Abstract

Take a one-parameter family of self-adjoint Fredholm operators [Formula: see text] on a Hilbert space [Formula: see text], joining endpoints [Formula: see text]. 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 [Formula: see text] acting in [Formula: see text], where [Formula: see text] denotes the multiplication operator [Formula: see text] for [Formula: see text]. Most results are about the case where the operators [Formula: see text] have compact resolvent. In this article, we review what is known when these operators have some essential spectrum and describe some new results. Using the operators [Formula: see text], [Formula: see text], an abstract trace formula for Fredholm operators with essential spectrum was proved in [23], extending a result of Pushnitski [35], although, still under strong hypotheses on [Formula: see text]: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text]. Associated to the pairs [Formula: see text] and [Formula: see text] are Krein spectral shift functions [Formula: see text] and [Formula: see text], respectively. From the trace formula, it was shown that there is a second, Pushnitski-type, formula: [Formula: see text] This can be employed to establish the desired equality, [Formula: see text] This equality was generalized to non-Fredholm operators in [14] in the form [Formula: see text] replacing the Fredholm index on the left-hand side by the Witten index of [Formula: see text] and [Formula: see text] on the right-hand side by an appropriate arithmetic mean (assuming [Formula: see text] is a right and left Lebesgue point for [Formula: see text] denoted by [Formula: see text] and [Formula: see text], respectively). But this applies only under the restrictive assumption that the endpoint [Formula: see text] is a relatively trace class perturbation of [Formula: see text] (ruling out general differential operators). In addition to reviewing this previous work, we describe in this article some extensions using a [Formula: see text]-dimensional setup, where [Formula: see text] are non-Fredholm differential operators. By a careful analysis we prove, for a class of examples, that the preceding trace formula still holds in this more general situation. Then we prove that the Pushnitski-type formula for spectral shift functions also holds and this then gives the equality of spectral shift functions in the form [Formula: see text] for the [Formula: see text]-dimensional model operator at hand. This shows that neither the relatively trace class perturbation assumption nor the Fredholm assumption are required if one works with spectral shift functions. The results support the view that the spectral shift function should be a replacement for the spectral flow in certain non-Fredholm situations and also point the way to the study of higher-dimensional cases. We discuss the connection with summability questions in Fredholm modules in an appendix.