Double operator integral methods applied to continuity of spectral shift functions

  • Alan L. Carey

    The Australian National University, Canberra, Australia
  • Fritz Gesztesy

    Baylor University, Waco, USA
  • Galina Levitina

    University of New South Wales, Sydney, Australia
  • Roger Nichols

    The University of Tennessee at Chattanooga, USA
  • Denis Potapov

    University of New South Wales, Sydney, Australia
  • Fedor Sukochev

    University of New South Wales, Sydney, Australia

Abstract

We derive two principal results in this note. To describe the first, assume that , , , , , are self-adjoint operators in a complex, separable Hilbert space , and suppose that

and

for some . Fix , odd, , and assume that for all ,

Then for any function in the class (cf. (1.1) for details),

Moreover, for each , , we prove the existence of constants and such that

which permits the use of differences of higher powers of resolvents to control the -norm of the left-hand side for .

Our second result is concerned with the continuity of spectral shift functions associated with a pair of self-adjoint operators in with respect to the operator parameter . For brevity, we only describe one of the consequences of our continuity results. Assume that and are fixed self-adjoint operators in , and there exists , odd, such that, , . For self-adjoint in we denote by the set of all self-adjoint operators in for which the containment , , holds. Suppose that and let denote a continuous path (in a suitable topology on , cf. (1.3)) from to in . If , then

The fact that higher powers , , of resolvents are involved, permits applications of this circle of ideas to elliptic partial differential operators in , . The methods employed in this note rest on double operator integral (DOI) techniques.

Cite this article

Alan L. Carey, Fritz Gesztesy, Galina Levitina, Roger Nichols, Denis Potapov, Fedor Sukochev, Double operator integral methods applied to continuity of spectral shift functions. J. Spectr. Theory 6 (2016), no. 4, pp. 747–779

DOI 10.4171/JST/140