Double operator integral methods applied to continuity of spectral shift functions

We derive two main results: First, assume that $A$, $B$, $A_n$, $B_n$ are self-adjoint operators in the Hilbert space $\mathcal{H}$, and suppose that $A_n$ converges to $A$ and $B_n$ to $B$ in strong resolvent sense as $n \to \infty$\mathrm{.}Fix$m \in $\mathbb{N}$$,$m$odd,$p \in [1,\infty)$,andassumethat$T:= \big[( A + iI_{$\mathcal{H}$})^{-m} - ( B + iI_{$\mathcal{H}$})^{-m}\big] \in $\mathcal{B}$p($\mathcal{H}$)$,$T_n := \big[( A_n + iI{$\mathcal{H}$})^{-m} - ( B_n + iI_{$\mathcal{H}$})^{-m}\big] \in $\mathcal{B}$p($\mathcal{H}$)$,and$\lim{n \rightarrow \infty} |T_n - T|{$\mathcal{B}$p($\mathcal{H}$)} =0$. Then for any function $f$ in the class $\mathfrak F{k}($\mathbb{R}$) \supset C_0^{\infty}($\mathbb{R}$)$(cf\mathrm{.}(1.1)),$$ \lim{n \rightarrow \infty} \big| [f(A_n) - f(B_n)] - [f(A)- f(B)]\big|{$\mathcal{B}$p($\mathcal{H}$)}=0. Our second result concerns the continuity of spectral shift functions $\xi(\cdot; B,B_0)$ with respect to the operator parameter $B$. For $T$ self-adjoint in $\mathcal{H}$ we denote by $\Gamma_m(T)$, $m \in \mathbb{N}$ odd, the set of all self-adjoint operators $S$ in $\mathcal{H}$ satisfying $\big[(S - z I_{\mathcal{H}})^{-m} - (T - z I_{\mathcal{H}})^{-m}\big] \in \mathcal{B}_1(\mathcal{H})$, $z \in \mathbb{C}\backslash \mathbb{R}$. Employing a suitable topology on $\Gamma_m(T)$ (cf. (1.9), we prove the following: Suppose that $B_1\in \Gamma_m(B_0)$ and let $\{B_{\tau}\}_{\tau\in [0,1]}\subset \Gamma_m(B_0)$ denote a path from $B_0$ to $B_1$ in $\Gamma_m(B_0)$ depending continuously on $\tau\in [0,1]$ with respect to the topology on $\Gamma_m(B_0)$. If $f \in L^{\infty}(\mathbb{R})$, then \lim{\tau\to 0^+} |\xi(, \cdot , ; B{\tau}, A_0) f - \xi(, \cdot , ; B_0, A_0) f|_{L^1($\mathbb{R}$; (|\nu|^{m+1} + 1)^{-1}d\nu)} = 0. $$