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On the index of a non-Fredholm model operator

Journal Article


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Abstract


  • Let {A(t)}t¿¿ be a path of self-adjoint Fredholm operators in a Hilbert space H, joining endpoints A± as t ¿ ±¿. Computing the index of the operator DA = ¿/¿t + A acting on L2(¿;H), where A denotes the multiplication operator (Af)(t) = A(t) f (t) for f ¿ L2(¿;H), 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. Introducing the operators H1 = D¿ADA and H2 = DAD¿A, we consider spectral shift functions, denoted by ¿(·;A+,A¿) and ¿ (·;H2,H1) associated with the pairs (A+,A¿) and (H2,H1). Under the restrictive hypotheses that A+ is a relatively trace class perturbation of A¿, a relationship between these spectral shift functions was proved in [14], for certain operators A± with essential spectrum, extending a result of Pushnitski [22]. Moreover, assuming A± to be Fredholm, the value ¿ (0;A¿,A+) was shown to represent the spectral flow along the path {A(t)}t¿¿while that of ¿ (0+;H1,H2) yields the Fredholm index of DA. The fact, proved in [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. In this situation it asserts that the Witten index of DA, denoted by Wr(DA), a substitute for the Fredholm index in the absence of the Fredholm property of DA, is given by (Formula Presented). Here one assumes that ¿ (·;A¿,A+) possesses a right and left Lebesgue point at 0 denoted by ¿L(0±;A+,A¿) (and similarly for ¿L(0+;H2,H1)). When the path {A(t)}t¿¿ 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.

Authors


  •   Gesztesy, Fritz (external author)
  •   Levitina, Galina (external author)
  •   Sukochev, F A. (external author)
  •   Carey, Alan L.

Publication Date


  • 2016

Citation


  • Carey, A., Gesztesy, F., Levitina, G. & Sukochev, F. (2016). On the index of a non-Fredholm model operator. Operators and Matrices, 10 (4), 881-914.

Scopus Eid


  • 2-s2.0-85006247016

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=7565&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/6535

Has Global Citation Frequency


Number Of Pages


  • 33

Start Page


  • 881

End Page


  • 914

Volume


  • 10

Issue


  • 4

Abstract


  • Let {A(t)}t¿¿ be a path of self-adjoint Fredholm operators in a Hilbert space H, joining endpoints A± as t ¿ ±¿. Computing the index of the operator DA = ¿/¿t + A acting on L2(¿;H), where A denotes the multiplication operator (Af)(t) = A(t) f (t) for f ¿ L2(¿;H), 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. Introducing the operators H1 = D¿ADA and H2 = DAD¿A, we consider spectral shift functions, denoted by ¿(·;A+,A¿) and ¿ (·;H2,H1) associated with the pairs (A+,A¿) and (H2,H1). Under the restrictive hypotheses that A+ is a relatively trace class perturbation of A¿, a relationship between these spectral shift functions was proved in [14], for certain operators A± with essential spectrum, extending a result of Pushnitski [22]. Moreover, assuming A± to be Fredholm, the value ¿ (0;A¿,A+) was shown to represent the spectral flow along the path {A(t)}t¿¿while that of ¿ (0+;H1,H2) yields the Fredholm index of DA. The fact, proved in [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. In this situation it asserts that the Witten index of DA, denoted by Wr(DA), a substitute for the Fredholm index in the absence of the Fredholm property of DA, is given by (Formula Presented). Here one assumes that ¿ (·;A¿,A+) possesses a right and left Lebesgue point at 0 denoted by ¿L(0±;A+,A¿) (and similarly for ¿L(0+;H2,H1)). When the path {A(t)}t¿¿ 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.

Authors


  •   Gesztesy, Fritz (external author)
  •   Levitina, Galina (external author)
  •   Sukochev, F A. (external author)
  •   Carey, Alan L.

Publication Date


  • 2016

Citation


  • Carey, A., Gesztesy, F., Levitina, G. & Sukochev, F. (2016). On the index of a non-Fredholm model operator. Operators and Matrices, 10 (4), 881-914.

Scopus Eid


  • 2-s2.0-85006247016

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=7565&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/6535

Has Global Citation Frequency


Number Of Pages


  • 33

Start Page


  • 881

End Page


  • 914

Volume


  • 10

Issue


  • 4