Double operator integral methods applied to continuity of spectral shift functions

Abstract

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 $T:=[(A+i{I}_{\mathcal{H}}{)}^{-m}-(B+i{I}_{\mathcal{H}}{)}^{-m}]\in {\mathcal{B}}_p(\mathcal{H})$, $T_n:=[(A_n+i{I}_{\mathcal{H}}{)}^{-m}-(B_n+i{I}_{\mathcal{H}}{)}^{-m}]\in {\mathcal{B}}_p(\mathcal{H})$, and ${\lim }_{n\to \infty }\Vert T_n-T{\Vert }_{{\mathcal{B}}_p(\mathcal{H})}=0$. Then for any function $f$ in the class ${\mathfrak{F}}_k(\mathbb{R})\supset C_0^{\infty }(\mathbb{R})$ (cf. (1.1)), $$\underset{n\to \infty }{\lim }\Vert [f(A_n)-f(B_n)]-[f(A)-f(B)]{\Vert }_{{\mathcal{B}}_p(\mathcal{H})}=0.$$ 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 $[(S-z{I}_{\mathcal{H}}{)}^{-m}-(T-z{I}_{\mathcal{H}}{)}^{-m}]\in {\mathcal{B}}_1(\mathcal{H})$, $z\in \mathbb{C}\backslash \mathbb{R}$. 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 $$\underset{\tau \to 0^{+}}{\lim }\Vert \xi (\cdot ;B_{\tau },A_0)f-\xi (\cdot ;B_0,A_0)f{\Vert }_{L^1(\mathbb{R};(|\nu {|}^{m+1}+1{)}^{-1}d\nu )}=0.$$

23 pages; we fixed an error in the original Theorem 2.12 which lead to considerable changes throughout this manuscript