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 $B_1in Gamma_m(B_0)$ and let ${B_{tau}}_{tauin [0,1]}subset Gamma_m(B_0)$ denote a path from $B_0$ to $B_1$ in $Gamma_m(B_0)$ depending continuously on $tauin [0,1]$ with respect to the topology on $Gamma_m(B_0)$. If $f in L^{infty}(mathbb{R})$, then $$ lim_{tauto 0^+} |xi(, cdot , ; B_{tau}, A_0) f - xi(, cdot , ; B_0, A_0) f|_{L^1(mathbb{R}; (| u|^{m+1} + 1)^{-1}d u)} = 0. $$
Download