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Double operator integral methods applied to continuity of spectral shift functions

Journal Article


Abstract


  • We derive two principal results in this note. To describe the first, assume that A, B, An, Bn, n ¿ N, are self-adjoint operators in a complex, separable Hilbert space H, and suppose that s-lim n¿¿(An - z0IH)-1 = (A - z0IH) -1 and s-lim n¿¿ (Bn - z0IH)-1 D (B - z0IH)-1 for some z0 ¿ C/R. Fix m ¿ N, m odd, p ¿ [1,¿), and assume that for all a ¿ R/{0}, (Equation presented). Then for any function f in the class Fm(R) ¿ C0¿.(R) (cf. (1.1) for details), lim n¿¿ [f (An) - f (Bn). - [f (A) - f (B)] Bp(H) = 0: Moreover, for each f ¿ Fm(R), p ¿ [1,¿), we prove the existence of constants a1; a2 ¿ R/(0) and C = C(f,m, a1, a2) ¿ (0,¿) such that (Equation presented), which permits the use of differences of higher powers m ¿ N of resolvents to control the · Bp(H)-norm of the left-hand side [f (A) - f (B) for f ¿ Fm(R). Our second result is concerned with the continuity of spectral shift functions ¿ (B;B0) associated with a pair of self-adjoint operators (B,B0) in H with respect to the operator parameter B. For brevity, we only describe one of the consequences of our continuity results. Assume that A0 and B0 are fixed self-adjoint operators in H, and there exists m ¿ N, m odd, such that, [(B0 - zIH)-m - (A0 - zIH)-m ¿ B1(H), z ¿ C/R. For T self-adjoint in H we denote by ¿m(T) the set of all self-adjoint operators S in H for which the containment [(S - zIH)-m - (T - zIH)-m ¿ B1(H), z ¿ C/R, holds. Suppose that B1 ¿ ¿m(B0) and let 1B o 2OE0;1 m.B0/ denote a continuous path (in a suitable topology on ¿m(B0), cf. (1.3)) from B0 to B1 in ¿m(B0). If f ¿ L¿(R), then (Equation presented) The fact that higher powers m 2 N, m ¿ 2, of resolvents are involved, permits applications of this circle of ideas to elliptic partial differential operators in Rn, n ¿ N. The methods employed in this note rest on double operator integral (DOI) techniques.

Authors


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

Publication Date


  • 2016

Citation


  • Carey, A., Gesztesy, F., Levitina, G., Nichols, R., Potapov, D. & Sukochev, F. (2016). Double operator integral methods applied to continuity of spectral shift functions. Journal of Spectral Theory, 6 (4), 747-779.

Scopus Eid


  • 2-s2.0-85006409404

Has Global Citation Frequency


Number Of Pages


  • 32

Start Page


  • 747

End Page


  • 779

Volume


  • 6

Issue


  • 4

Abstract


  • We derive two principal results in this note. To describe the first, assume that A, B, An, Bn, n ¿ N, are self-adjoint operators in a complex, separable Hilbert space H, and suppose that s-lim n¿¿(An - z0IH)-1 = (A - z0IH) -1 and s-lim n¿¿ (Bn - z0IH)-1 D (B - z0IH)-1 for some z0 ¿ C/R. Fix m ¿ N, m odd, p ¿ [1,¿), and assume that for all a ¿ R/{0}, (Equation presented). Then for any function f in the class Fm(R) ¿ C0¿.(R) (cf. (1.1) for details), lim n¿¿ [f (An) - f (Bn). - [f (A) - f (B)] Bp(H) = 0: Moreover, for each f ¿ Fm(R), p ¿ [1,¿), we prove the existence of constants a1; a2 ¿ R/(0) and C = C(f,m, a1, a2) ¿ (0,¿) such that (Equation presented), which permits the use of differences of higher powers m ¿ N of resolvents to control the · Bp(H)-norm of the left-hand side [f (A) - f (B) for f ¿ Fm(R). Our second result is concerned with the continuity of spectral shift functions ¿ (B;B0) associated with a pair of self-adjoint operators (B,B0) in H with respect to the operator parameter B. For brevity, we only describe one of the consequences of our continuity results. Assume that A0 and B0 are fixed self-adjoint operators in H, and there exists m ¿ N, m odd, such that, [(B0 - zIH)-m - (A0 - zIH)-m ¿ B1(H), z ¿ C/R. For T self-adjoint in H we denote by ¿m(T) the set of all self-adjoint operators S in H for which the containment [(S - zIH)-m - (T - zIH)-m ¿ B1(H), z ¿ C/R, holds. Suppose that B1 ¿ ¿m(B0) and let 1B o 2OE0;1 m.B0/ denote a continuous path (in a suitable topology on ¿m(B0), cf. (1.3)) from B0 to B1 in ¿m(B0). If f ¿ L¿(R), then (Equation presented) The fact that higher powers m 2 N, m ¿ 2, of resolvents are involved, permits applications of this circle of ideas to elliptic partial differential operators in Rn, n ¿ N. The methods employed in this note rest on double operator integral (DOI) techniques.

Authors


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

Publication Date


  • 2016

Citation


  • Carey, A., Gesztesy, F., Levitina, G., Nichols, R., Potapov, D. & Sukochev, F. (2016). Double operator integral methods applied to continuity of spectral shift functions. Journal of Spectral Theory, 6 (4), 747-779.

Scopus Eid


  • 2-s2.0-85006409404

Has Global Citation Frequency


Number Of Pages


  • 32

Start Page


  • 747

End Page


  • 779

Volume


  • 6

Issue


  • 4