Double operator integral methods applied to continuity of spectral shift functions

We derive two principal results in this note. To describe the first, assume that A, B, A_n, B_n, n ϵ N, are self-adjoint operators in a complex, separable Hilbert space H, and suppose that s-lim n∞→(A_n - z₀IH)^-1 = (A - z₀IH) ^-1 and s-lim n∞→ (B_n - z₀IH)^-1 D (B - z₀IH)^-1 for some z₀ ϵ C/R. Fix m ϵ N, m odd, p ϵ [1,∞), and assume that for all a ϵ R/{0}, (Equation presented). Then for any function f in the class F_m(R) ∪ C₀∞.(R) (cf. (1.1) for details), lim n∞→ [f (A_n) - f (B_n). - [f (A) - f (B)] B_p(H) = 0: Moreover, for each f ϵ F_m(R), p ϵ [1,∞), we prove the existence of constants a₁; a₂ ϵ R/(0) and C = C(f,m, a₁, a₂) ϵ (0,∞) such that (Equation presented), which permits the use of differences of higher powers m ϵ N of resolvents to control the · B_p(H)-norm of the left-hand side [f (A) - f (B) for f ϵ F_m(R). Our second result is concerned with the continuity of spectral shift functions ϵ (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 ϵ N, m odd, such that, [(B₀ - zIH)^-m - (A₀ - zIH)^-m ϵ B₁(H), z ϵ C/R. For T self-adjoint in H we denote by Λ_m(T) the set of all self-adjoint operators S in H for which the containment [(S - zIH)^-m - (T - zIH)^-m ϵ B₁(H), z ϵ C/R, holds. Suppose that B₁ ϵ Λ_m(B₀) and let 1B o 2OE0;1 m.B0/ denote a continuous path (in a suitable topology on Λ_m(B₀), cf. (1.3)) from B₀ to B₁ in Λ_m(B₀). If f ϵ 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 ϵ N. The methods employed in this note rest on double operator integral (DOI) techniques.