On the index of a non-Fredholm model operator

Abstract

, and its relation to spectral flow along this path, has a long history, but it is mostly focussed on the case where the operators A(t) all have purely discrete spectrum. [14], that these values of the two spectral functions are equal, resolves the index = spectral flow question in this case. This relationship between spectral shift functions was generalized to non-Fredholm operators in [9] again under the relatively trace class perturbation hypothesis. Introducing the operators H H H H In this situation it asserts that the Witten index of D D D D A A A A , denoted by W r (D D D D A A A A ) , a substitute for the Fredholm index in the absence of the Fredholm property of D D D D A A A A , is given byHere one assumes that ξ ( ·;A − ,A + ) possesses a right and left Lebesgue point at 0 denoted by ξ L (0 ± ;A + ,A − ) (and similarly for ξ L (0 + ;H H H H 2 ,H H H H 1 ) ).When the path {A(t)} t∈R consists of differential operators, the relatively trace class perturbation assumption is violated. The simplest assumption that applies (to differential operators in 1+1 dimensions) is to admit relatively Hilbert-Schmidt perturbations. This is not just an incremental improvement. In fact, the method we employ here to make this extension is of interest in any dimension. Moreover we consider A ± which are not necessarily Fredholm and we establish that the relationships between the two spectral shift functions found in all of the previous papers [9] , [14], and [22], can be proved, even in the non-Fredholm case. The significance of our new methods is that, besides being simpler, they also allow a wide class of examples such as pseudodifferential operators in higher dimensions. Most importantly, we prove the above formula for the Witten index in the most general circumstances to date.Mathematics subject classification (2010): Primary 47A53, 58J30; Secondary 47A10, 47A40.

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

“…However [11] alone is not enough to establish a trace formula of the kind stated in our abstract for the model operator (1.5). Nor can we use [14], where non-Fredholm operators were studied, as it needs assumption (iii ′ ) as well and so cannot be applied to our present context.…”

“…Remark 1.1 shows that one has, for the examples we study in this paper, a relatively Hilbert-Schmidt perturbation condition. This Hilbert-Schmidt constraint also appears in [11] in an abstract setting where it is used to obtain a Pushnitskitype formula.…”

“…This motivated the paper [11] which seeks to provide an abstract framework for generalizations of the Pushnitski results.…”

“…While this progression, going from the (0 + 1)-dimensional case in [23] to the (1 + 1)-dimensional case both here and in [11] may appear incremental at first sight, approximation methods in [11] and [12] to make progress on the general case of model operators in higher dimensions are in preparation. Hence our interest in giving here an accessible exposition via a class of models of this new approach.…”

“…⋄ Thus, in differential operator terms, [14,23] considered the zero-dimensional case because, if we write in the notation of the remark, A + = A − + F , then F (1 + A 2 − ) −1/2 = (A + − A − )(1 + A 2 − ) −1/2 which is trace class only if n = 0. In this article, following the ideas in [11], we consider the situation when (A + − A − )(1 + A 2 − ) −1/2 is Hilbert-Schmidt and show how this allows some general one-dimensional examples. We note that our results point the way to an attack on the problem of partial differential operators in higher dimensions.…”