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 $n \to \infty$. Fix $m \in \mathbb{N}$, $m$ odd, $p \in [1,\infty)$, and assume that $T:= \big[( A + iI_{\mathcal{H}})^{-m} - ( B + iI_{\mathcal{H}})^{-m}\big] \in \mathcal{B}_p(\mathcal{H})$, $T_n := \big[( A_n + iI_{\mathcal{H}})^{-m} - ( B_n + iI_{\mathcal{H}})^{-m}\big] \in \mathcal{B}_p(\mathcal{H})$, 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. (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 $$\lim_{\tau\to 0^+} \|\xi(\, \cdot \, ; B_{\tau}, A_0) f - \xi(\, \cdot \, ; B_0, A_0) f\|_{L^1(\mathbb{R}; (|\nu|^{m+1} + 1)^{-1}d\nu)} = 0.$$